The Catholic Apologist Dave Armstrong explains this mystery by analogy to the Incarnation. He says (in his book, A Biblical Defense of Catholicism) that the relationship of the elements before and after their transubstantiation is like the union of deity and humanity in the Person of Christ. The Catholic Encyclopedia also explains the doctrine in incarnational terms. I do not know whether that makes it official dogma, but I am responding to it as such, until shown otherwise.
That analogy presents a serious problem, even worse that the error of transubstantiation itself.
I agree with the Catholic Church that Christ is one Person, uniting in Himself the deity of the Second Person of the Trinity and the humanity born of the Virgin Mary. We also agree that these two natures are distinct, though not separate, not mixed or confused, but each retaining its respective nature. This is formulated in the Chalcedonian Creed. For me, the official statement is found in the Westminster Confession of Faith VIII:2, "So
that two whole, perfect, and distinct natures, the Godhead and the manhood,
were inseparably joined together in one person, without conversion, composition,
In contrast, in the Fifth Century, a man named Eutyches argued that the Person of Christ was of one nature, which was both divine and human.This doctrine is known as Eutychianism, in his honor, or Monophysitism. It is the official doctrine of the Coptic, Armenian, and Syriac churches, but is rejected by Catholic, Orthodox, and Protestant churches, as a perversion of the Person of Christ, in that it turns Him, not into God or man, but rather into a hybrid which is neither fully one or the other. This was the historical situation which resulted in the Council of Chalcedon in 451.
Do you see why I bring this up? If Jesus is fully human, as both I and the Catholic Church affirm, then He has a body with the essential attributes of any other human being. That includes two things, off the top of my head, which are relevant here: first, that He can only be physically in one place at a time, and second, that His flesh is itself, neither transformable into something else, nor subject to transformation from something else. One aspect of that is the question of where Jesus now is. So, where does the Bible say that is? Heaven (Luke 24:51 and Rom. 8:34)!
The Church of Rome, on the other hand, claims that the flesh of Christ is wherever His deity is, which is, of course, everywhere! Is that not the teaching of Eutyches? Is that not what was rejected by Chalcedon, almost sixteen centuries ago? Yes, it is! The Catholic doctrine of transubstantiation intermixes the humanity and divinity of Christ, so that each shares in the attributes of the other.
Rome claims that their doctrine is from "tradition," yet it wasn't formally adopted until the Fourth Lateran Council in 1215. Thus, the orthodox doctrine which was expressed in the Fifth Century was denied by Rome in the Thirteenth. That, to my mind, puts the lie to Catholic claims of tradition, not that tradition would have justified a doctrine of such superstitious and idolatrous implications. It also indicates that no person, claiming the Name of Christ and housing the Holy Spirit in his heart, can participate in a Catholic eucharist, as if it were the same thing as true communion in the body and blood of Christ.