At the end of any turn where Paris falls to the Axis, do the following:
(1) For each territory and sea zone containing French pieces, roll a die.
(2) On a result of 1-3, the territory and units stay aligned with the Allies, as Free French.
(3) On a result of 4-6, the territory and units become pro-Axis neutrals, the Vichy French.
Use German counters to indicate which units and territories are Vichy (these behave just as pro-Axis neutrals).
This is a simple way to model this historical effect. Likewise,
At the end of any turn where Rome falls to the Allies, do the following:
(1) For each territory and sea zone containing Italian pieces, roll a die.
(2) On a result of 1-3, the territory and units stay aligned with the Axis, as the Italian Socialist Republic.
(3) On a result of 4-6, the territory and units become pro-Allied neutrals, the Co-Belligerent Italian forces.
Use US counters to indicate which units and territories are Co-Belligerent (these behave just as pro-Allied neutrals).
If desired, you could extend this as a general house rule to apply to all powers that lose their capitals.
I predict that it would result in some very interesting game-states…
Happy home-brewing, M_I_R