A quick Google search reliably informed me that she’s “bidialectal”, dahling. Yes, this word really does exist! Or at least it does Stateside…I couldn’t find a definition in the Oxford Dictionary online, but Merriam Webster came up trumps:
BIDIALECTALISM: facility in using two dialects of the same language (also: the teaching of Standard English to pupils who normally use a nonstandard dialect).
It seems most bidialectal celebrities, Gillian Anderson included, moved from one English-speaking country to another at a young age, and adopted their new (now perfect) accent in an attempt to fit in.
Perhaps the most bizarre example of a bidialectal actor is John Barrowman, who was born in Glasgow to Scottish parents but grew up in Illinois. Here’s a compilation of clips showing how he can quite naturally flit between the two (it’s interesting but I suggest you only watch the first few minutes – whoever put this film together is clearly a massive John Barrowman fan).