Here is an idea about A Vichy France rule. When Paris was taken by Germany during WWII, Most of France joined the Vichy regime. François Darlan and the navy remained under Vichy along with many colonies. Normandy-Bordeaux and Paris was German territories and southern France the Vichy capital.
French Indochina was lost for the Vichy early in the war along with French equatorial Africa and Syria.
Morocco, Algeria,Tunisia, French west Africa, French central Africa, some part of French equatorial Africa, French Guyana and Madagascar remained more or less under Vichy control. But all joined Free France during the war except for the northern Africa countries that remained under Vichy until the allied invasion.
How about when Paris is taken Normandy-Bordeaux, Southern France, the navy in Mediterranean and the french force in North Africa joined the Nazis, but half of the units in France is moved to UK represents Free French forces who fled and the rest of the French territories in Africa and around the world remains unde Free French control. Â