Today we celebrate the feast of Corpus Christi, to commemorate the institution of the Holy Eucharist. Pope Benedict's comments on unity with respect to this feast can be found here.
In his Epistle to the Smyrnaeans, St. Ignatius (d. 107 AD) says the following of certain heretics:
They abstain from the Eucharist and from prayer because they confess not the Eucharist to be the flesh of our Saviour Jesus Christ, which suffered for our sins, and which the Father, of His goodness, raised up again. Those, therefore, who speak against this gift of God, incur death in the midst of their disputes. But it were better for them to treat it with respect, that they also might rise again.
To which heretics is St. Ignatius referring? Docetists. We can see that from what he says only a few paragraphs before.
"But if these things were done by our Lord only in appearance, then am I also only in appearance bound. And why have I also surrendered myself to death, to fire, to the sword, to the wildbeasts? ... For what does any one profit me, if he commends me, but blasphemes my Lord, not confessing that He was [truly] possessed of a body?" (my emphases)
In discussions with various Evangelicals, I quite commonly encounter the following sort of claim:

In January of this year, I argued in "The Incarnation and Church Unity" that this notion [i.e. that the unity of the Church is invisible] is a form of ecclesial docetism. It treats the Body of Christ as fundamentally immaterial, spiritual, and invisible, having only a visible appearance in the world, but not actually being a visible Body. Why do I say that it treats the Body of Christ as though it is not actually a visible Body? Because visible unity is essential and intrinsic to a body; if a body ceases to be visibly one, it ceases to be. So if visible unity is only accidental to something, that something is not a living body; it is, at most, only the appearance of a body. Hence those who claim that the Body of Christ is invisibly one and visibly divided, are treating the Body of Christ as though it were merely an *apparent* Body, not an actual Body. And therefore it should be clear why this position is rightly described as ecclesial docetism.
In response to my argument, Jonathan Bonomo suggested that Catholic ecclesiology is Eutychian. He wrote:
[I]f we're going to compare our ecclesiologies to Christological heresies, I don't see how yours would escape the charge of Eutychianism: mixing the inner/outer until they become prone to confusion. I'm not myself making the charge, mind you. But I don't see why making a distinction between the inner and outer while ardently holding to their essential union should be charged with docetism, while essentially identifying them should be exempt from other corresponding charges.
Must one choose between Docetic and Eutychian ecclesiologies? Is the Catholic ecclesiology Eutychian? No and No. Docetism denies that Christ ever took on a human nature. According to Docetism, Christ only appeared to be human, but did not actually have a human body. Eutychianism, which is also called Monophysitism (meaning "one nature") was condemned at the Fourth General Council, the Council of Chalcedon in 451 AD. According to the Monophysites, Jesus' humanity was absorbed into His divine nature such that He no longer has a human nature, having only His divine nature (hence "Monophysitism").
Docetism and Eutychianism both deny that Christ has a human nature. For that reason, both Docetic and Eutychian notions of the Church (i.e. the Body of Christ) treat the Church as in itself invisible, spiritual, and immaterial, only visible in the sense that it makes use of embodied human believers in much the same way that the Logos (i.e. the Second Person of the Trinity), according to a Docetic conception, might make use of material elements in order to appear as though having a body, but would not actually be made up of those material elements, nor would they be parts of Him. Chalcedonian Christology, with its affirmation of two distinct natures united without mixture in one hypostatic union, entails that the Church as the Body of Christ is in itself visible and hierarchically organized as one corporate entity. (cf. Mystici Corporis Christi, 16) The real distinction between Christ's divine nature and His human nature does not imply that the Body of Christ is not necessarily visibly one. Rather, it is His having a real human nature that entails that the Body of Christ is necessarily visibly one.
Jonathan apparently thinks that the Catholic claim [that the visible Body of Christ is essentially one] mistakenly attributes to the visible aspect of the Church what is only true of the invisible aspect of the Church, and in that way falsely attributes what is only true of the divine nature to the human nature, as Eutychianism does. But Jonathan's claim is based on the implicit assumption that a living human body is not essentially visibly one. And yet a living human body is essentially visibly one. If it ceases to be visibly one, it ceases to be. Hence, its visible unity is essential to its being. Therefore, the Catholic claim that the [visible] Body of Christ is essentially visibly one is not Eutychian. Rather, the charge that Catholic ecclesiology is Eutychian is based on the mistaken notion that visible unity is not intrinsically essential to a living human body.
One possible objection is that the Eucharistic Body of Christ is not essentially visibly one, since there are many consecrated hosts. And so therefore the Mystical Body of Christ is not necessarily essentially visibly one. But the Eucharistic Body of Christ differs from the Mystical Body of Christ in an important and relevant way. It is not the case that a consecrated host is a *part* or *member* of the Eucharistic Body of Christ.
Christ is present whole and entire in each of the species and whole and entire in each of their parts, in such a way that the breaking of the bread does not divide Christ. (CCC 1377)
But Christ is the Head of His Mystical Body. Every member of the Mystical Body of Christ is joined to Christ, but only He is the Head. That means that Christ is not present "whole and entire in" the members of the Mystical Body of Christ in the way He is in each of the parts of the Eucharistic species. The members of the Mystical Body of Christ must remain united to the Head of that Body, in order to remain in that Body, but no part of a consecrated Host must remain visibly united to the other parts of that Host, in order to remain wholly and entirely Christ.
Christ and his Church thus together make up the "whole Christ" (Christus totus). (CCC 795)
For that reason, the Mystical Body is essentially visibly one, even though "the breaking of the bread does not divide Christ".
To see video clips of our Corpus Christi procession and benedictions from last year, see here, here, here, here, and here. Thanks to Mark Abeln of Rome of the West for these videos.