Spoiler ahead! Though of an Italian novel...
There's an AH novel in Italian in which Italy remains neutral well into the war, and then it is attacked by Germany simply because it is neutral but Allied-friendly.
The Germans never manage to fully occupy Italy, even though they make use of Vichy French (now German-allied), Croatian, Hungarian etc. troops to bolster their numbers. They do cross the Alps, but at a heavy price, then they advance South but get delayed at multiple fortification lines along rivers and the Apennines (reminds us of something...).
In a nice twist, there is an Italian civil war - but Mussolini and his regime remain loyal to the Allies, while it's the King and his old-guard generals that change sides. This doesn't do much good to the Germans.
Not invading Italy means that air bases in Italy can be used for a bomber campaign against Southern Germany, as well as Ploesti.
I'll add that in 1937, Italy was producing 10% of the total world production of bauxite. Yugoslavia 9%. Germany OTOH was producing 2%. Even capturing a 16% from France (not from colonies, thus within german reach) and a 13% from Hungary, that's a big chunk of aluminium - aircraft fuselages - that becomes unavailable to the Axis, unless the Axis come and take it.