And how do we know she's not having darker hair in the Eyrie? Have there been any set photos of Sophie at the Eyrie? I've only seen the ones from Dubrovnik/KL.
We don't know for sure, but she never changed her hair at any point during filming, so the only way the hair change is happening is if it's a wig (she also, for instance, had red hair in Iceland when filming Vale stuff, so again, wig is the only option). Which could be the case, for all we know, but considering Turner's hair is dyed red to begin with, I don't know why they wouldn't just dye it again.