I think my problem is solved. The 7riii apparently uses the modelling lights (tungsten in my case) to create the Auto White Balance, and not the actual light from the flash unit when the photo is taken. Choosing the Flash WB setting doesn't improve matters either.
The solution is to find the Custom White Balance setting in the menus, where the camera will ask you to point the camera at a white card, press the central button of the Rear Control Wheel, and all is then well, in theory.
And in practice, things are indeed much much better now.