A plectic Taniyama group

We revisit Langlands' construction of the Taniyama group to define a plectic Taniyama group for any totally real field F. The construction is functorial in F and recovers Langlands' original Taniyama group when F = Q. We relate our construction to Nekovar's 'hidden symmetries' in classical CM theory, generalising the relationship between the Taniyama group and Tate's half-transfer maps.