Garry Wills is the author of “Why I am a Catholic,” but also of “Why Priests?” and “Papal Sins: Structures of Deceit.”  His “Lincoln at Gettysburg” won a Pulitzer Prize.  He also wrote a biography of Augustine, St. Augustine (a Penguin Lives Biography).  So, it might be good for folks to pay attention when he says (Why Priests, p. 16):

Indeed, Eucharist (“Thanksgiving”) in its later sense, of sharing bread and wine as the body and blood of Christ, is never used in the New Testament, not even in the Letter to Hebrews, which alone calls Jesus a priest. Even when the term “Eucharist” came in, as with the letters of Ignatius of Antioch, it was still, as in Paul, simply a celebration of the people’s oneness at the “one altar.” That meaning for the “body of Christ” would persist as late as the fourth and fifth centuries, in Augustine’s denial of the real presence of Jesus in the elements of the meal.

What you see passes away, but what is invisibly symbolized does not pass away. It perdures. The visible is received, eaten, and digested. But can the body of Christ be digested? Can the church of Christ be digested? Can Christ’s limbs be digested? Of course not. [[Augustine, Sermon 227]]

If you want to know what is the body of Christ, hear what the Apostle [Paul] tells believers: “You are Christ’s body, and his limbs” [1 Cor 12.27]. If, then, you are Christ’s body and his limbs, it is your symbol that lies on the Lord’s altar–what you receive is a symbol of yourselves. When you say “Amen,” and you must be the body of Christ to make that “Amen” take effect. And why are you bread? Hear again the Apostle, speaking of this very symbol: “We are one bread, one body, many as we are” [1 Cor 10.17].[[Augustine, Sermon 272]]

Believers recognize the body of Christ when they take care to be the body of Christ. They should be the body of Christ if they want to draw life from the spirit of Christ. No life comes to the body of Christ but from the spirit of Christ.[[Augustine, In Joannem Tractatus 26.13]]