Dominus Iesus:
“The Catholic faithful are required to profess that there is an historical continuity — rooted in the apostolic succession53 — between the Church founded by Christ and the Catholic Church: “This is the single Church of Christ… which our Saviour, after his resurrection, entrusted to Peter’s pastoral care (cf. Jn 21:17), commissioning him and the other Apostles to extend and rule her (cf. Mt 28:18ff.), erected for all ages as ‘the pillar and mainstay of the truth’ ( 1 Tim 3:15). This Church, constituted and organized as a society in the present world, subsists in [ subsistit in ] the Catholic Church, governed by the Successor of Peter and by the Bishops in communion with him”.54 With the expression subsistit in, the Second Vatican Council sought to harmonize two doctrinal statements: on the one hand, that the Church of Christ, despite the divisions which exist among Christians, continues to exist fully only in the Catholic Church, and on the other hand, that “outside of her structure, many elements can be found of sanctification and truth”,55 that is, in those Churches and ecclesial communities which are not yet in full communion with the Catholic Church.56 But with respect to these, it needs to be stated that “they derive their efficacy from the very fullness of grace and truth entrusted to the Catholic Church”.57 17. Therefore, there exists a single Church of Christ, which subsists in the Catholic Church, governed by the Successor of Peter and by the Bishops in communion with him.58 The Churches which, while not existing in perfect communion with the Catholic Church, remain united to her by means of the closest bonds, that is, by apostolic succession and a valid Eucharist, are true particular Churches.59 Therefore, the Church of Christ is present and operative also in these Churches, even though they lack full communion with the Catholic Church, since they do not accept the Catholic doctrine of the Primacy, which, according to the will of God, the Bishop of Rome objectively has and exercises over the entire Church.”
-end of Dominus Iesus quotation-
Branch theory can not be accepted by Eastern or Western Catholicism, whether it includes Anglicans or only those with true Apostolic Succession, or anyone else. If Branch Theory is true, then any bishop who does not like anything in his Church can just leave and establish his own True Full Church. How far do we go? Are heretics with Apostolic Succession part of True Church or not? Are only those in schism? Both Orthodox and Catholic theologians put it nicely, that Apostolic teachings on Schism disregard this theory. Hence it can not be supported by anyone in communion with any of those Churches.