Once you have Sardinia, Corsica has no hope.
I don't believe the Axis can hold anywhere against determined Allied assault forever. The question is how much can you make them pay? How much to make fighting for it a 'win' even after the islands are lost?
Between bombing out the 4 ports, and having naval control from Sardinia, you have no chance of holding Corsica.
How late would you consider obtaining Corsica and Sardinia to be a 'problem'?
Plus I retarget my 6th TF on turn one to hit Corsica later if I need it too. If I do land that way, Corsica would be in ruins without hope of supply.
How many divisions would you plan to send to liberate Sardinia and Corsica?
How many task forces would you devote to the task?
Do you plan to land on them at a specific time period?
How do you plan to respond if after the Italian surrender Corsica and Sardinia are just a black hole?
Sardinia is just as easily taken, as it has only 4 ports too. If you can't get supplies, you can't hold them. Island hopping 101.
I can't hold them forever, but how long of me holding them creates problems for what else you want to accomplish in the Med?
If it is Dec '43 and you have 10-20 divisions, with 2-5 TFs, committed to clearing Sardinia, and I still own half the island with little more than three divisions, who is getting the strategic win?
It's too easy for the Allies to bomb out all the ports. Once that happens, nothing heavy can get in or out; at that point the Germans have to rely on airlifts. While workable, that could get very expensive in transport planes, and any heavy equipment will be left behind. Airlift is unreliable, expensive, and messy.
Bombers devoted to trying to close ports (which is actually not easy considering they get automatic priority repairs and having a handful of engineers available isn't difficult - they help with the digging beforehand anyway) aren't bombing the Reich.
I setup most of my defenses in the first two turns. I usually see the depots in Sardinia fully stocked before Italy surrenders. Being on the defensive isn't very costly overall in supplies, and German units eat lighter.
I can see merit to putting a Regt in every port on Corsica; this would lock out the Italians, and prevent the Allies from easy support. I would keep Airlift nearby for a quick extraction. If the Allies show up on even one beach, though, it would be time to bug out for me. Germans could drop a Fort Zone in each port, but a fort zone by itself could be taken by the Italians.
Check the Italian surrender rules. The Germans have additional units at their disposal to help counter the Italian Problem. Security units can also serve to draw out the island liberation afterwards, forcing battles for every hex, and possibly getting reserve backstop from 'real' combat units.
It may also be possible for the Allies to counter that Corsica move by airlifting in troops, and supplying by air; tricky, but if the Island is isolated by sea I think the Allies can probably take it that way. They just need 1 para Div to take a port, and it's probably over for Germany. Allies should have access to at least one airfield when Italy surrenders.
Security units are very light to air transport, and they hardly require any supplies. They can generate decent defensive CV in FZs on rough terrain if the digging has already been done. Enough to thwart a lone infantry division, especially if one of the aforementioned 'real' combat regiments activates from reserve.
If someone tries to liberate a port with 1 airborne division they're going to become POWs.