I am realising, as I reflect deep and wide on this matter, that potentially I do not have time to explore every nook and cranny of the dark cave called "Anglican claims to consistency" so what I offer here will have a number of shortcomings. Perhaps down the track I can seek to overcome them. Here goes for now.
I see Anglicans claiming consistency for their theologies resting their claims on these presuppositions (one or more of them): the Spirit at work in our midst guides and leads us (charismatically through individuals; collectively through meetings); Scripture teaches us thus and so (and where we change our minds on the basis of Scripture we may have a new understanding of what Scripture really means); Sixteenth century Anglicanism (i.e. as represented in the BCP, the Thirty-Nine Articles, and Hooker's Ecclesiastical Polity) lays a foundation which we consistently follow). Any other hand "S" words to consider?
Thus we might find a 21st century "progressive Anglican" claiming on the one hand that a new decision is "the Spirit at work in our midst" and "the consequence of applying reason (cf. Hooker)" to the matter.
A 21st century "evangelical Anglican" might emphasise "Scripture teaches thus and so" with a reserve argument, "and it is in keeping with Article A, B, or C."
Then a 21st century "catholic Anglican" might draw on the Spirit, Scripture and the Sixteenth century to support a proposal.
The ordination of women to the presbyterate or episcopacy is a case in point: the Spirit has moved the church to do this, Scripture (rightly interpreted) does not forbid it, indeed offers positive examples of female apostolate, and it stands to reason. (With opposition claiming Scripture does not allow it and the Sixteenth century did not provide for it).
There are varieties within these broad schools of Anglican thinking. Some catholic Anglicans feel more comfortable with making decisions in keeping with the great catholic churches (Rome, Eastern Orthodox) so it is not a question of being against the ordination of women (per se) but of when the other churches would also approve this development; other catholic Anglicans would see themselves in the vanguard of discerning the Spirit's lead on the matter and look forward to the day when other churches catch them up.
Within evangelical Anglicanism there are those who are 'neo Puritan' in the sense that they see Sixteenth century Anglicanism as a bit of a muddle: the Puritan vision of reformed Anglicanism was stymied then and subsequently, which is a pity, because Anglicanism has been lacking consistency ever since. The 21st century is a new opportunity to complete the Puritan reforms.
My next and probably last post on this theme will look at the following questions: on the one hand, is it consistent to support the ordination of women and not to support the blessing of same sex relationships (both, arguably, departures from Scripture and from catholic doctrine), and is it consistent to oppose lay presidency at the eucharist while supporting the ordination of women (arguably the former is not prohibited by Scripture and latter is; while both the former and the latter are prohibited by catholic doctrine)?