Linking AppleCare plan from one country with Apple ID registered in another country
I am a fairly recent immigrant to France and I have retained my UK Apple ID while also registering an Apple ID in France in order to install a few apps that are only available there.
I have just purchased an iPhone in France and have taken out AppleCare on the monthly plan. Do I have to link the AppleCare with my French Apple ID?
If so does that mean I have to use my French AppleID as my primary ID on my iPhone? I would rather not do that but link it instead to my UK Apple ID.
Thanks
iPhone 15 Pro Max