What they're saying is they want a licence or certificate (in the same way that you can get a US licence issued on the basis of a foreign PPL) issued in the same nationality of the aircraft being flown, or some form of paperwork conferring the privileges by that state, for you to use those licence privileges in a foreign aircraft.
So, what you need is a a UK licence with IR for a 'G' reg aircraft, French licence for an 'F'.
Now - if you had an 'N' reg aircraft you'd have no problems, License and aircraft being of the same state.