The Oxford Dictionary of the Christian Church defines the branch theory as:
…the theory that, though the Church may have fallen into schism within itself and its several provinces or groups of provinces be out of communion with each other, each may yet be a branch of the one Church of Christ, provided that it continues to hold the faith of the original undivided Church and to maintain the Apostolic Succession of its bishops. Such, it is contended by many Anglican theologians, is the condition of the Church at the present time, there being now three main branches…
William Palmer (1803–1885), an Oxford theologian, was the principal originator of the Branch Theory. His two-volume Treatise on the Church of Christ (1838) formulated the notion. The theory was then popularized during the Oxford Movement particularly through the work of the Tractarians. However, some leaders of the movement became unsatisfied and later converted to Roman Catholicism.
The Roman Catholic Church accepts the theory's two fundamental premises, namely that maintaining the teachings of the ancient Christian Church is essential and that apostolic succession is sufficient for valid orders within another Church despite being in a state of schism or heresy. However, it does not accept that those separated by schism or heresy are part of the one Church. Moreover it deems that Anglican orders are in general invalid, and that, while individual Anglicans may have orthodox faith, the Anglican Church has not maintained the fulness of ancient Christian teachings, most notably on the sacraments.
John Paul II used the "two-lungs" metaphor, holding that, for its full health, the Roman Catholic Church needs union with the Churches of the Orient, in the same way that the human body is not fully healthy unless it is breathing with both lungs.
Eastern Orthodoxy does not have a central authority, but there is an equality of bishops with structures based on mere honorable precedence. Thus, there is a wide range of opinions. Ecumenical Patriarch Bartholomew I of Constantinople, who is considered primus inter pares of Eastern Orthodoxy, agrees with the branch theory. For example, when he received PopeBenedict XVI spoke about relations between Eastern Orthodoxy and Roman Catholicism he considered them to be separated branches of the One Church of Jesus Christ. .
The Eastern Orthodox Church has traditionally recognized the validity of Roman Catholic sacraments such as baptism, eucharist and ordination, and does not re-baptize or re-ordain laymen and clergy into its ranks.
Another position, frequently referred to as antiecumenism rejects the Branch Theory as being incompatible with the nature of the Church and therefore an ecclesiological heresy. An example for that is this statement put forth by the 1983 Holy Synod of Bishops of the Russian Orthodox Church Outside Russia:
Those who attack the Church of Christ by teaching that Christ's Church is divided into so-called "branches" which differ in doctrine and way of life, or that the Church does not exist visibly, but will be formed in the future when all "branches" or sects or denominations, and even religions will be united into one body; and who do not distinguish the priesthood and mysteries of the Church from those of the heretics, but say that the baptism and eucharist of heretics is effectual for salvation; therefore, to those who knowingly have communion with these aforementioned heretics or who advocate, disseminate, or defend their new heresy of Ecumenism under the pretext of brotherly love or the supposed unification of separated Christians, Anathema!
The above position can be summarized in what they perceive the Anglican communion to be; namely a heresy within a heretical Roman Catholic Church.
With the exception of a few Lutheran bodies the great majority of Protestant churches do not consider themselves apostolic in the sense of a literal apostolic succession. Therefore, such churches do not accept the fundamental premises of the Branch Theory which, in and of itself, excludes them.