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Anglicanism was coined to describe the common religious tradition of these churches; {as also that of }the Scottish Episcopal Church, which, though originating earlier within the Church of Scotland, had come to be recognised as sharing this common identity.

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It's an abbreviated reference, to save having to say the same thing twice...

Anglicanism was coined to describe the common religious tradition of these churches; as also [(to describe) the common religious tradition] of the Scottish Episcopal Church

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that = relative (anaphoric) pronoun referring back to an earlier noun or noun-phrase

as = in the same way, similarly

also = in addition, as well, too

and also too that [i.e. the common religious tradition] of the Scottish Episcopal Church

Compare:

This "Renaissance" of the Latin language (as also that of ancient Greek) did not start any earlier than the 8th-9th century.

as also the "Renaissance" of ancient Greek...

and the "Renaissance" of ancient Greek as well

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