The Eucharist, Christ's Body and Blood, revealed to us as the Real Presence, is to be honored and worshipped as the Sign of God in the World. We cannot confuse the Eucharist with the Mystical Body of Christ, which is the Church. Christ is the Head of the Church and we are the Members. He is Present to us in the Eucharist and the Indwelling of the Holy Trinity, which we receive in baptism.
To state that the Presence of Christ is the Body of Christ is to forget that He is the Head, and we the Members. Some of us are feet, some legs, some hands and so forth according to our gifts. The Church, created by Christ on earth, as an institution which is Holy, One, Apostolic, and Catholic, is a sign of Christ's Presence in the world for all ages.
The Eucharist is the Lord and the Lord causes the unity.
St. Paul explains that the Mystical Body of Christ is made up of those of us who are baptized and in the Church. Here is a section from Mystici Corporis, so that we can look at the definition of the Body of Christ as present and the Presence of the Eucharist. Remember that we belong to a visible Church not an invisible one. The Mystical Body of Christ is the Catholic Church and not the Christians outside of it. That is clear in the second section here.
14. That the Church is a body is frequently asserted in the Sacred Scriptures. "Christ," says the Apostle, "is the Head of the Body of the Church." If the Church is a body, it must be an unbroken unity, according to those words of Paul: "Though many we are one body in Christ." But it is not enough that the Body of the Church should be an unbroken unity; it must also be something definite and perceptible to the senses as Our predecessor of happy memory, Leo XIII, in his Encyclical Satis Cognitum asserts: "the Church is visible because she is a body. Hence they err in a matter of divine truth, who imagine the Church to be invisible, intangible, a something merely "pneumatological" as they say, by which many Christian communities, though they differ from each other in their profession of faith, are untied by an invisible bond.
15. But a body calls also for a multiplicity of members, which are linked together in such a way as to help one another. And as in the body when one member suffers, all the other members share its pain, and the healthy members come to the assistance of the ailing, so in the Church the individual members do not live for themselves alone, but also help their fellows, and all work in mutual collaboration for the common comfort and for the more perfect building up of the whole Body.
16. Again, as in nature a body is not formed by any haphazard grouping of members but must be constituted of organs, that is of members, that have not the same function and are arranged in due order; so for this reason above all the Church is called a body, that it is constituted by the coalescence of structurally untied parts, and that it has a variety of members reciprocally dependent. It is thus the Apostle describes the Church when he writes: "As in one body we have many members, but all the members have not the same office: so we being many are one body in Christ, and everyone members one of another." 
22. Actually only those are to be included as members of the Church who have been baptized and profess the true faith, and who have not been so unfortunate as to separate themselves from the unity of the Body, or been excluded by legitimate authority for grave faults committed. "For in one spirit" says the Apostle, "were we all baptized into one Body, whether Jews or Gentiles, whether bond or free." As therefore in the true Christian community there is only one Body, one Spirit, one Lord, and one Baptism, so there can be only one faith. And therefore, if a man refuse to hear the Church, let him be considered - so the Lord commands - as a heathen and a publican.  It follows that those who are divided in faith or government cannot be living in the unity of such a Body, nor can they be living the life of its one Divine Spirit.
40. But we must not think that He rules only in a hidden  or extraordinary manner. On the contrary, our Redeemer also governs His Mystical Body in a visible and normal way through His Vicar on earth. You know, Venerable Brethren, that after He had ruled the "little flock"  Himself during His mortal pilgrimage, Christ our Lord, when about to leave this world and return to the Father, entrusted to the Chief of the Apostles the visible government of the entire community He had founded. Since He was all wise He could not leave the body of the Church He had founded as a human society without a visible head. Nor against this may one argue that the primacy of jurisdiction established in the Church gives such a Mystical Body two heads. For Peter in view of his primacy is only Christ's Vicar; so that there is only one chief Head of this Body, namely Christ, who never ceases Himself to guide the Church invisibly, though at the same time He rules it visibly, through him who is His representative on earth. After His glorious Ascension into Heaven this Church rested not on Him alone, but on Peter, too, its visible foundation stone. That Christ and His Vicar constitute one only Head is the solemn teaching of Our predecessor of immortal memory Boniface VIII in the Apostolic Letter Unam Sanctam;  and his successors have never ceased to repeat the same.
41. They, therefore, walk in the path of dangerous error who believe that they can accept Christ as the Head of the Church, while not adhering loyally to His Vicar on earth. They have taken away the visible head, broken the visible bonds of unity and left the Mystical Body of the Redeemer so obscured and so maimed, that those who are seeking the haven of eternal salvation can neither see it nor find it
65. For this reason We deplore and condemn the pernicious error of those who dream of an imaginary Church, a kind of society that finds its origin and growth in charity, to which, somewhat contemptuously, they oppose another, which they call juridical. But this distinction which they introduce is false: for they fail to understand that the reason which led our Divine Redeemer to give to the community of man He founded the constitution of a Society, perfect of its kind and containing all the juridical and social elements - namely, that He might perpetuate on earth the saving work of Redemption, - was also the reason why He willed it to be enriched with the heavenly gifts of the Paraclete. The Eternal Father indeed willed it to be the "kingdom of the Son of his predilection;" but it was to be a real kingdom in which all believers should make Him the entire offering of their intellect and will, and humbly and obediently model themselves on Him, Who for our sake "was made obedient unto death." There can, then, be no real opposition or conflict between the invisible mission of the Holy spirit and the juridical commission of Ruler and Teacher received from Christ, since they mutually complement and perfect each other - as do the body and soul in man - and proceed from our one Redeemer who not only said as He breathed on the Apostles "Receive ye the Holy Spirit," but also clearly commanded: "As the Father hath sent me, I also send you;" and again: "He that heareth you, heareth me."
and most importantly, it is the Indwelling of the Holy Spirit which brings about this unity
78. For indeed We are not ignorant of the fact that his profound truth - of our union with the Divine Redeemer and in particular of the indwelling of the Holy spirit in our souls - is shrouded in darkness by many a veil that impedes our power to understand and explain it, both because of the hidden nature of the doctrine itself, and of the limitations of our human intellect. But We know, too, that from well-directed and earnest study of this doctrine, and from the clash of diverse opinions and the discussion thereof, provided that these are regulated by the love of truth and by due submission to the Church, much light will be gained, which, in its turn will help to progress in kindred sacred sciences. Hence, We do not censure those who in various ways, and with diverse reasonings make every effort to understand and to clarify the mystery of this our wonderful union with Christ. But let all agree uncompromisingly on this, if they would not err from truth and from the orthodox teaching of the Church: to reject every kind of mystic union by which the faithful of Christ should in any way pass beyond the sphere of creatures and wrongly enter the divine, were it only to the extent of appropriating to themselves as their own but one single attribute of the eternal Godhead. And, moreover, let all hold this as certain truth, that all these activities are common to the most Blessed Trinity, insofar as they have God as supreme efficient cause.
And here comes the point overlooked entirely by the author of the said pamphlet, which caused me to write these posts. Christ is the Head of the Body. We are the Body. One cannot say that the community of the Church equals the Presence of Christ the Head. Christ's Presence in the Church is that of the Bridegroom loving His Bride. The community is not some sort of cosmic Christ. God dwells within us through the sacraments and in that unity of the Indwelling of the Holy Trinity, we become one with God. We DO NOT become God.
81. It seems to Us that something would be lacking to what We have thus far proposed concerning the close union of the Mystical Body of Jesus Christ with its Head, were We not to add here a few words on the Holy Eucharist, by which this union during this mortal life reaches, as it were, a culmination.
82. By means of the Eucharistic Sacrifice Christ our Lord willed to give the faithful a striking manifestation of our union among ourselves and with our divine Head, wonderful as it is and beyond all praise. For in this Sacrifice the sacred minister acts as the viceregent not only of our Savior but of the whole Mystical Body and of each one of the faithful. In this act of Sacrifice through the hands of the priest, by whose word alone the Immaculate Lamb is present on the altar, the faithful themselves, united with him in prayer and desire, offer to the Eternal Father a most acceptable victim of praise and propitiation for the needs of the whole Church. And as the Divine Redeemer, when dying on the Cross, offered Himself to the Eternal Father as Head of the whole human race, so "in this clean oblation" He offers to the heavenly Father not only Himself as Head of the Church, but in Himself His mystical members also, since He holds them all, even those who are weak and ailing, in His most loving Heart.
83. The Sacrament of the Eucharist is itself a striking and wonderful figure of the unity of the Church, if we consider how in the bread to be consecrated many grains go to form one whole, and that in it the very Author of supernatural grace is given to us, so that through Him we may receive the spirit of charity in which we are bidden to live now no longer our own life but the life of Christ, and to love the Redeemer Himself in all the members of His social Body.
84. As then in the sad and anxious times through which we are passing there are many who cling so firmly to Christ the Lord hidden beneath the Eucharistic veils that neither tribulation, nor distress, nor famine, nor nakedness, nor danger, nor persecution, nor the sword can separate them from His love, surely no doubt can remain that Holy Communion which once again in God's providence is much more frequented even from early childhood, may become a source of that fortitude which not infrequently makes Christians into heroes.
85. If the faithful, Venerable Brethren, in a spirit of sincere piety understand these things accurately and hold to them steadfastly, they will the more easily avoid those errors which arise from an irresponsible investigation of this difficult matter, such as some have made not without seriously endangering Catholic faith and disturbing the peace of souls.
86. For there are some who neglect the fact that the Apostle Paul has used metaphorical language in speaking of this doctrine, and failing to distinguish as they should the precise and proper meaning of the terms the physical body, the social body, and the Mystical Body, arrive at a distorted idea of unity. They make the Divine Redeemer and the members of the Church coalesce in one physical person, and while they bestow divine attributes on man, they make Christ our Lord subject to error and to human inclination to evil. But Catholic faith and the writings of the holy Fathers reject such false teaching as impious and sacrilegious; and to the mind of the Apostle of the Gentiles it is equally abhorrent, for although he brings Christ and His Mystical Body into a wonderfully intimate union, he nevertheless distinguishes one from the other as Bridegroom from Bride.
To distinguish between the Presence of Christ among His People and not to say that He is His People is a clear split with many post-Vatican II commentators.
And we love this Body:
92. But lest we be deceived by the angel of darkness who transforms himself into an angel of light, let this be the supreme law of our love: to love the Spouse of Christ as Christ willed her to be, and as He purchased her with His blood. Hence, not only should we cherish exceedingly the Sacraments with which holy Mother Church sustains our life, the solemn ceremonies which she celebrates for our solace and our joy, the sacred chant and the liturgical rites by which she lifts our minds up to heaven, but also the sacramentals and all those exercises of piety by which she consoles the hearts of the faithful and sweetly imbues them with the Spirit of Christ. As her children, it is our duty, not only to make a return to her for her maternal goodness to us, but also to respect the authority which she has received from Christ in virtue of which she brings into captivity our understanding unto the obedience of Christ. Thus we are commanded to obey her laws and her moral precepts, even if at times they are difficult to our fallen nature; to bring our rebellious body into subjection through voluntary mortification; and at times we are warned to abstain even from harmless pleasures. Nor does it suffice to love this Mystical Body for the glory of its divine Head and for its heavenly gifts; we must love it with an effective love as it appears in this our mortal flesh - made up, that is, of weak human elements, even though at times they are little fitted to the place which they occupy in this venerable body.
And for the Church Militant................................
106. Moreover, Christ proved His love for His spotless Bride not only at the cost of immense labor and constant prayer, but by His sorrows and His sufferings which He willingly and lovingly endured for her sake. "Having loved His own...He loved them unto the end." Indeed it was only at the price of His Blood that He purchased the Church. Let us then follow gladly in the bloodstained footsteps of our King, for this is necessary to ensure our salvation: "For if we have been planted together in the likeness of His Resurrection." and "if we be dead with him, we shall live also with Him." Also our zealous love for the Church demands it, and our brotherly love for the souls she brings forth to Christ. For although our Savior's cruel passion and death merited for His Church an infinite treasure of graces, God's inscrutable providence has decreed that these graces should not be granted to us all at once; but their greater or lesser abundance will depend in no small part on our own good works, which draw down on the souls of men a rain of heavenly gifts freely bestowed by God. These heavenly gifts will surely flow more abundantly if we not only pray fervently to God, especially by participating every day if possible in the Eucharistic Sacrifice; if we not only try to relieve the distress of the needy and of the sick by works of Christian charity, but if we also set our hearts on the good things of eternity rather than on the passing things of this world; if we restrain this mortal body by voluntary mortification, denying it what is forbidden, and by forcing it to do what is hard and distasteful; and finally, if we humbly accept as from God's hands the burdens and sorrows of this present life. Thus, according to the Apostle, "we shall fill up those things that are wanting of the sufferings of Christ in our flesh for His Body, which is the Church."
This encyclical was written in 1943, at the blackest times of WWII. We can identify, today, with the problems, torments and even persecutions against the Body of Christ, for which we pray.
109. Before concluding, We cannot refrain from again and again exhorting all to love holy Mother Church with a devoted and active love. If we have really at heart the salvation of the whole human family, purchased by the precious Blood, we must offer every day to the Eternal Father our prayers, works and sufferings, for her safety and for her continued and ever more fruitful increase. And while the skies are heavy with storm clouds, and exceeding great dangers threaten the whole of human Society and the Church herself, let us commit ourselves and all that we have to the Father of Mercies, crying out: "Look down, we beseech Thee, Lord, on this Thy family, for which our Lord Jesus Christ did not hesitate to be betrayed into the hands of evil men and to undergo the torment of the Cross."