What is "The Catholic Church"?
In order to answer this question, a fine quote from Archbishop Mark Haverland's text "Anglican Catholic Faith and Practice," available from the Anglican Parishes Association
“The Catholic Church is the one, true Church of God throughout the world. Anglicans reject the idea that the Roman Catholic Church is the Catholic Church. The Catholic Church is wherever Christians gather around a bishop of apostolic succession and faith. The presence of Apostolic Succession and faith, and thus Catholicity, certainly requires and presupposes, acceptance of the Old and New Testaments as the record of God’s decisive self-revelation; acceptance of the doctrine of the undivided, ancient Church, which is summarized particularly in the Creeds, celebration of the sacraments, particularly in Baptism and the Eucharist, through which God unites his people to himself; maintenance of the historical line of bishops sometimes called the tactile succession. Mere maintenance of a mere outward or tactile line of succession does not by itself maintain catholicity: the faith and worship of the Church also must be maintained.”
I've added the italics for emphasis. I think this is an important point to take note of, in that sometimes Anglicans place episcopacy over faith. The point, the purpose of the episcopacy is to transmit the Christian faith inviolate. If bishops cease to do this they break the catholicity of the Church.