I have Catholic monarchist friends who have no problem getting along with fellow Catholics who have no use for Monarchy whatsoever. Ultimately, their religion comes first. The same is probably true in Orthodoxy, though I'm less familiar with those internal debates. That would frankly be more difficult for me. (Thankfully I don't seem to encounter anti-monarchist Anglicans very often.)
I suppose the fundamental difference between me and those who are firmly committed to the exclusive truth of their particular variety of Christianity is this: while I understand that a Christian must put Christ first (that is, while there are non-Christian monarchies I like, I wouldn't defend Saudi Arabia from justified Christian criticism simply because it calls itself a Kingdom), I'm not willing to put the divisions _within_ Christianity ahead of Monarchism. I would be Catholic if I lived in France in 1685, Protestant if I lived in England in 1570, and Orthodox if I lived in Tsarist Russia, but I would not become Muslim if I lived in Constantinople in 1453...I hope. So, I'm a Christian first, but a Monarchist second, and an Anglican/Protestant (if Anglicanism is Protestant, but that's a whole other discussion) third. To the extent that I identify as an American at all, that's fourth at best.