The former "branch theory", which was primarily a belief of the "high church" party in the Anglican Communion, was that the Church consisted of the Roman-centered Church, the Orthodox, and the Church of England. It has been years since I have heard it mentioned as the break-up of the Anglican Communion rendered it both moot and obviously indefensible.

Given what has happened to that Communion, it is perplexing that at least some Catholics would like to repeat the experiment in the Catholic Church. Fortunately a strong teaching authority and the guidance and protection of the Holy Spirit will keep the barque afloat to the end of time.