From the diocese of new Westminster:
The Anglican Communion described in 1930 at the Lambeth Conference: “…a fellowship, within the one Holy Catholic and Apostolic Church, of those duly constituted dioceses, provinces or regional Churches in communion with the See [Diocese] of Canterbury…”
In other words, “dioceses, provinces, or regional Churches” are in the Anglican Communion if they are “duly constituted” and Canterbury wants to be in communion with them.
In practice, how you can tell whether you’re still on the good side of the See of Canterbury seems to work out as being invited by the Archbishop of Canterbury to the Lambeth Conference.
It’s up to the Archbishop of Canterbury, the Most Rev. Rowan Williams, to decide who to invite. Up till now at least, the Archbishops of Canterbury have invited everyone, except in very rare instances of competing bishops in the same geographical area.
There’s nothing said about heads of the various national Anglican Churches, the Primates, helping the Archbishop of Canterbury decide who’s to come – although the present Archbishop of Canterbury in some statements seems to have suggested he might seek advice.
To turn to history, it was North Americans who got us into this strange situation in the first place.
It was an American bishop from Vermont who originally had the idea of a Lambeth Conference. But it was Canadian Bishops, who in 1865 urgently asked for the then 144 bishops in the Anglican Communion to meet at Lambeth in 1867.
Read it all. Please note that the author fails to give adequate attention to the way in which the controversy focused on how to work with and understand Holy Scripture. As Bishop Colenso wrote:
the Pentateuch, as a whole, cannot personally have been written by Moses, or by anyone acquainted personally with the facts which it professes to describe, and, further, that the (so-called) Mosaic narrative, by whomsoever written, and though imparting to us, as I fully believe it does, revelations of the Divine Will and Character, cannot be regarded as historically true.