If it's fair to even call it "communion."
If we want to understand the political theology of the American founding and its attendant religious liberty and establishment issues, we need to understand the dynamic of how The Church of England (Anglicans) dealt with the separation.
The "official rules" of the Church of England held that the Monarch was head of both Church and State. The top clerical official is the Archbishop of Canterbury, who "reports" to the Monarch. If one did not affirm the Monarch's rightful place as leader of the Church, one could face severe legal penalties from both the civil magistrate as well as Church canons, up to and including excommunication.
I've noted before the irony that so many of America's leading Founding Fathers were Anglicans, and what they technically did was rebel against the head of their Church. If they were "Anglican fundamentalist" (high church types who followed every single rule of the C of E down to the letter), they would have been Tories and submitted to the King, because that's what the Church officially taught.
But even in Mother England, high church Anglicanism of the "fundamentalist" variety wasn't the only game in town in the C of E, even if perhaps it prevailed. Even King George III, about whose personal religious convictions I'm not exactly sure, I seriously doubt was an "Anglican fundamentalist" (even though that theology benefited his self interest). I'm assuming "the Christian King" was some kind of orthodox Trinitarian Anglican (?); but the attitude of the Monarchy towards America, up until things got heated with their dispute seemed to be one of (as the late Sen. Daniel Patrick Moynihan put it in an entirely different context) "benign neglect."
The variety of Anglicanism that appealed to the Whig Patriots of the American founding was that of "low church latitudinarianism." Latitudinarianism literally means "doctrinal latitude." Now, most of these latitudinarians were probably "orthodox" on the Trinity and related doctrines; but not all of them. Or at least, their "doctrinal latitude" made room for more deistic and unitarian minded theists to feel comfortable in the Church.
If one wants a name of a latitudinarian figure that America's founders greatly respected, look up Bishop Benjamin Hoadly.
Over in America during the revolution, Bishop William White was concerned that the conflict would fracture the Church. And his concerns were valid. As a matter of technicality, the Church of England only had jurisdiction in England. If America is no longer England, then the Church of England no longer exists there, even if the buildings and believers remain. Many of the believers left. The revolution indeed gutted the C of E in America.
But when America successfully rebelled, the C of E in America, by necessity had to "start over." The Anglican hierarchy in England no longer had any power or jurisdiction over America. Ultimately what ended up happening was because Bishops White and Samuel Seabury (and other Anglican power players in America) were committed to historic Anglican orthodoxy, what emerged in American Episcopalianism was something traditional and orthodox.
It didn't have to be that way though. In New England, one of the Anglican Churches, King's Chapel went unitarian after the split. Indeed, if Bishop James Madison whom many suspected was heterodox, got his druthers and was in charge of rewriting the rules for Southern Anglicans and got his cousin and namesake and Thomas Jefferson to assist, we could have had a Unitarian Episcopalian system there too.
But what of the issue of "communion" among American and English Anglicans, post revolution? The first American Bishop was Samuel Seabury, a Tory loyalist and "The Farmer" whom Alexander Hamilton purported to "refute." He was jailed during the revolution for his loyalism. But after America won, he wished to remain and help rebuild the C of E in America, now as The Protestant Episcopal Church.
Seabury traveled to Great Britain to get consecrated by the C of E. But he ran into a problem. The then extant rules officially demanded he take an oath of loyalty to the crown. Seabury wisely refused because he knew that wouldn't fly in America. But he got consecrated anyway by the Scottish Episcopal Church, composed of non-juring Bishops who "borrowed" from the Church of England's theology, but without recognizing any of their authority.
So at that time, Seabury was America's first and only existing Bishop and was in communion with a church that was in schism with the Church of England. This turned out to be a wise and strong move on Seabury's part. Great Britain ended up changing its rules to accommodate America's new situation. They apparently did NOT want American Episcopalians to be in communion only with the schismatic Jacobite Church.
So they relented and consecrated the next three American Bishops, William White, Samuel Provost and James Madison. In the Church of England. And I'm assuming without the "loyalty to the crown" oaths.