How do advocates of transubstantiation understand 1 Corinthians 10:16?

In 1 Corinthians 10:16, the Apostle Paul says that: > The cup of blessing which we bless, is it not a participation in the blood of Christ? The bread which we break, is it not a participation in the body of Christ? (ESV) Transubstantiation theology says that when the priest says the words of consecration, the substance, but not the accidents, of the bread and wine are changed to the body, blood, soul and divinity of Jesus. However, Paul identifies the Eucharist as participation bread and wine **and** body and blood. How is this reconciled? To clarify what exactly my question is, I need to make clear that I am not saying Paul is contradicting REAL PRESENCE theology. That is, I don’t think this verse in Paul’s epistle demonstrates that Christ is not physically present in the Eucharist. Instead, I am asking specifically about CATHOLIC real presence theology, which has transubstantiation built in. For a counter example, consider the general Eastern Orthodox and Lutheran views, which hold that bread and wine AND the body, blood, soul, and divinity are present in the communion meal.