The TAC is not a part of the Anglican Communion. It considered itself as being so until February 22nd 1994, which is presumably when the issue of womens' orders became irrevocable.
It has a strong attachment to the Book of Common Prayer and, at least in the US, permits no deviation from the 1928 form. In the UK, it permits the use of the English Missal—an English translation of the Extraordinary Form of the Mass.
In its St Louis Declaration (which is really the closest it comes to a formulation of doctrine) it professes belief in the seven sacraments, and the sacrificial nature of the Eucharist.
On the minus side, it lists the 39 articles as being amongst its authoritative documents. How it reconciles one or two of these articles with the last observation is beyond me. I was never convinced by Newman's Tract 90.
The TAC isn't a large operation in the UK: only twelve parishes in England, as far as I can tell, and I've not a clue as to how large these are. Some of them have websites: here, and here. I see that in the latter case, Letchworth, they have already canonized Henry VI themselves.
The Letchworth parish worships in a Liberal Catholic building, and has posted photographs of a High Mass there. I should have thought that the Liberal Catholics (who cheerfully ordain women) were very strange bedfellows for the TAC.
The Liberal Catholic Church has as one of its basic tenets freedom of thought. It "permits to lay members entire freedom in the interpretation of Creeds, Scriptures and Tradition, and of the Liturgy. The Church holds strongly that belief should be the result of individual study or intuition, not its antecedent. A truth is not a truth for a man, nor a revelation a Revelation, until he sees it to be true for himself." ("Statement of Principles")And I recognize the celebrant: he has attended various services that I have been involved with in the past.
In the end, all Christians who seek reconciliation with the Holy See have to be made welcome if they recognize its truth and desire communion. The 'Uniate Anglican' system in the States has worked very well, on the whole, and there seems little reasons why it should not do so in the UK. In the States, they willingly accepted a modification of the Book of Common Prayer, though it was a little weird—the Novus Ordo offertory was incorporated, with 'you' language instead of the 'thou' used elsewhere, and the Roman Canon in Miles Coverdale's 16th century translation—and as far as I know continue to use it, though some churches now celebrate the Extraordinary Form of the Roman Rite and others the Ordinary Form.
There is another issue, and perhaps my friend William might like to comment if he is reading this. The TAC, I'm sure, consists of worthy and saintly clergy and people. But as far as I can see, it is, well, fringe, to put it at its most charitable. All things considered, at least in the UK, it isn't a major player. However, there is a much larger group which is more mainstream, if I can put it like that, and this is the SSC; the Societas Sanctæ Crucis, or Society of the Holy Cross.
I have known a number of SSC members over the years, and have found them to be, largely, an admirable body of chaps. They are in communion (if uneasily) with the C of E, but are unquestionably at the 'Catholic' end, and nearly unanimous in their opposition to womens' orders. What is particularly impressive is that they have a real spirituality; there is a rule of life which they are expected to follow, there are retreats and days of recollection. It consists of both celibate and married clergy. They are organized into regional chapters which, for many of them, take the place of the 'official' liberal diocesan organization. Probably most of them account themselves under the authority and sacramental ministry of the 'Flying Bishops'. The majority use the Ordinary Form of the Roman Rite in their Eucharists.
If Rome is to accept the TAC into full Communion, then it should keep an eye also on the SSC. It would be a great shame if the TAC were to call all the shots and, effectively, queer the pitch for a much larger and more coherent (in both senses of the word) group who are increasingly interested (it seems to me) in making the same sort of move.
Perhaps more in later posts.