For some Anglicans, whether still in the Anglican Communion or somewhere in the Continuing spectrum of ecclesial bodies, the prospect of submission to the Pope is unpleasant; some instead look East.
The Nonjurors, unsurprisingly, entered into correspondence with the Eastern Orthodox Patriarchs back in the early 18th century; equally unsurprisingly, the Orthodox indicated that the Nonjurors, who represented themselves as the remnant of the ancient Catholic and Orthodox British Church (O Anglicanism! how Janus-faced thou art, shewing whatever visage will appear fairest to whomsoever thou courtest!), must nonetheless accept every single unchanged doctrine of the Holy Orthodox Church, including belief in the change in the bread and wine into the very Body and Blood of Christ with no bread and wine remaining save but in outward appearance (the Greek term used answers to the Latin transsubstantio), invocation of the saints, and veneration of images. These doctrines, plus the suggestion that they should give up their liturgy and use that of St John Chrysostom, the Nonjurors persisted in rejecting, and had the nerve to ask might be considered optional, "second order" issues (cf. Rowan!), which they could leave aside, and yet nonetheless be admitted to "inter-communion". Anglicans of such stiff-necked sorts don't change: all they really want is everyone else to acknowledge them right.
At the time, these negotiations came to nothing, for neither side would give way, and moreover the Patriarchs found out from the then Archbishop of Canterbury that these Nonjurors were an insignificant group of dissenters.
In the late 19th century, some American Episcopalians approached the Russian Orthodox Bishop Tikhon (later glorified by the Russian Church as a saint), asking to become Orthodox but - as usual - keep their Prayer Book. The Russian Church theologians compiled some very valuable notes on the American B.C.P., very perceptively noting its deficiencies and points wherein its Protestantism would have to be corrected. In the event, no one was reconciled at the time...
In past decades, groups of Western Orthodox have been at length finally established, typically using either a version of the Roman Mass Byzantinized with an inserted Epiclesis after the words of consecration, or a mixture of Anglican and some Roman prayers styled the Liturgy of St Tikhon. The latter in particular is a strange hybrid, a real dog's breakfast; I think it an example of how not to compose or correct a rite (e.g. in the amended Prayer of Consecration and Oblation taken from Anglican books, several clauses of the Roman Canon are slotted in, alongside an Epiclesis from Chrysostom's Divine Liturgy).
From what I can make out, these Uniates - for that is what they are, and the Orthodox should not be ashamed to acknowledge this word, for all that they condemn Rome for having Eastern Catholic jurisdictions - are fairly small in number, and some practise faintly ridiculous pastimes, such as composing Offices for obscure English saints, whose cultus is revived precisely because they luckily died ante 1054: for after that date, all were but Western heretics in their eyes.