1: Anglicanism and Congregationalism
The response of American conservatives in seeking schism has been to essentially jettison the traditional hallmarks of Angicanism--a hierarchical church led by bishops acting in concert within the national province, loosely affiliated in a world-wide Communion bound not by a curia but rather by ties of sympathy. That's gone now; American conservatives are demanding the right, on the parish and even on the diocesan level to make their own policy, and to secede from the national church, while remaining the recognized Anglican presence within their geograpgical limits.
See, e.g., http://www.virtueonline.org/portal/modules/news/article.php?storyid=6961;
At the same time, they grant the national church no such autonomy, hailing raiding forays from Nigeria and Uganda to try to displace the national church within parishes and even dioceses that have decided to bolt, and calling for Communion imposed discipline.
See, e.g., http://www.standfirminfaith.com/index.php/site/article/7052/
This is a pretty peculiar form of churchmanship from a movement that was born in the creation of a national established church in England, quite frankly. Moreover, it has long been established that the unity of the national church was a high priority as early as the organization of the Conventions of 1785-1789. Indeed, the dictates of unity transcended national borders in the founding period; notably a major objection to the consecration of Bp. Seabury was that, by receiving his orders from the Church of Scotland, he had been guilty of schismatic behavior toward the Church of England. (Manross, A History of the American Episcopal Church (1935) at 194; 195-202)). As a thesis significantly antedating the present troubles makes clear, secession was not deemed an option throughout the Church's history from the Civil War until the present era. See J.A. Dator, Confederal, Federal or Unitary (1959), which is archived at: http://www.edow.org/dator/ (Hat tip: the inestimable Fr. Jake!)
Moreover, while conservatives argue that parishes and dioceses are free to secede from the national church at whim, they also insist that the national church should be subject to discipline by the Anglican Communion as a whole, acting through the Primates. This innovation would create an Anglican Curia, but also note the odd structure that results: the Primates become, absent any formal act empowering them to do so, sovereign over national churces--which themselves enjoy no such sovereighty over the individual dioceses or parishes, under TEC's polity. In other words, the national church has accountability without authority, while the local parishes and individual dioceses enjoy autonomy without accountability.
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