I notice that France Fights On had them pull out of Corsica/be pushed out. Corsica was apparently too close to Italy to be able to defend properly with the limited air support available to the Allies at that point.
While the Italian navy wasn't a lot of threat, any time Allied ships tried to reinforce/support the island, they got hammered. Is my recollection of that TL.
Good point here. Once the Germans finish their invasion of France - August 15, 1940 - they obviously base aircrafts in Southern France. That, and Italy, there's no way Corsica survives for long.
Admittedly, Corsica is farther from mainland France than Great Britain was, but it is also a much smaller island, meaning even the southern tip is within reach of the Luftwaffe.
It is not a coincidence if the FFO German invasion of Corsica is called "operation Merkur"

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Because the Germans have no landing ships, they go the OTL Crete way: parachutists. They barely managed to establish a couple of bridgeheads.
After ferocious air and sea battles, the Italian navy and Luftwaffe units based in southern France ultimately prevails.
The french Air force can't defend Corsica airspace as much as it should because the Luftwaffe is fixing them with a so-call "Blitz against Tunis and Algiers". The AdA has to keep lot of fighter squadrons in North Africa for air defence, and that ultimately tip the balance in favour of the Luftwaffe over Corsica, saving the bridgeheads there.
Stukas also sunk the Bearn carrier, damaging french battleships and cruisers.
Note: before Corsica felt, there were talks between De Gaulle and Churchill about "basing Wellingtons [bombers] in Corsica" to bomb Italy. Corsica, incidentally, was Napoleon birth place. You can imagine De Gaulle reaction at Churchill suggestion...