As the bishop of the Episcopal Diocese of South Carolina draws new scrutiny at the invitation of some members of the diocese, folks are renewing their struggle with various and strained notions of what it is to be a member of the Anglican Communion. May I presume to try to help? It seems well and good to pay attention to Anglican ecclesiology here. Neither 'South Carolina' the state nor 'South Carolina' the Episcopal Diocese thereof has the ability to 'remain a member of the Anglican Communion.' Only the Churches of which dioceses are subsidiary parts are members of said Communion.
In addition, an 'appeal to the Archbishop of Canterbury' is completely outside the ecclesiology of both the Protestant Episcopal Church in the United States of America (PECUSA) and the Church of England. Both Churches are independent of one another. Neither the Church of England nor the Archbishop of Canterbury has ecclesiastical authority in the PECUSA. An appeal to Canterbury from a member of this Church in the 21st century is reminiscent of someone from the Church of England appealing to Rome in the late 16th century. In either case, it is entirely un-Anglican thinking to suppose that such is appropriate. This, besides the fact that it is unconstitutional and non-canonical.
It is quite accurate to note, as some have, that 'the Church in the USA was not a member of the Anglican Communion in 1785'. This is simply because there was no such thing as an 'Anglican Communion' in 1785. Churches on the Anglican model that existed at that time were three in number: the Church of England, the Scottish Episcopal Church, and the Protestant Episcopal Church in the United States of America.
There were no independent Anglican Churches beyond these because there were no independent nations beyond these who sustained an Anglican ecclesiology. It wasn't until England began losing its colonies that Churches independent of England began constituting themselves in these former colonies. With the increase in number of these independent Churches, the Church of England thought it beneficial to its waning influence to suggest a Lambeth Conference in the late 1800's.
Further, it wasn't until the late 1960's that the Anglican Communion as we know it today was convened in the form of the Anglican Consultative Council. It is, of course, a simplistic myth that the Archbishop of Canterbury declares, in a sort of ex cathedra manner, which Churches are and which are not members of the Anglican Communion. He has the prerogative to invite or not invite bishops to the Lambeth Conference, as we have seen in his shamefully public dis-invitation of the Rt. Rev. Gene Robinson a few years ago. But people will notice, I hope, that this had no affect whatsoever on whether or not the PECUSA remained a member of the Anglican Communion.
Thus, this Archbishop has himself demonstrated that neither an invitation to the Lambeth Conference or a lack thereof determines membership in the Anglican Communion. Membership is determined by appeal and petition to the Anglican Consultative Council and by the Council's subsequent vote in response. Inasmuch as the Council is an inter-Church body, the Church of England's Archbishop of Canterbury has but a small role in the process. And again, membership in the Anglican Communion is for Churches, not dioceses. There is no avenue for subsidiary part of a Church, such as the Episcopal Diocese of South Carolina or any other Episcopal diocese, to have unilateral membership in the Anglican Communion. I hope this is helpful.