what if at the end of Spanish Civil War the Italians REALLY do attempt to occupy the Balearic Islands?
http://en.wikipedia.org/wiki/Italian_occupation_of_Majorca
(similar to their occupation of Dodecanese at the other end of the Med
http://en.wikipedia.org/wiki/Italian_Islands_of_the_Aegean)
Italy annexed the Dodecanese as sovereign territory after defeating Turkey in a war. Italy cannot do that to Spain. Italy can just seize the Balearics, and Spain cannot resist, but it would be an extremely ugly thing for Italy to do.
Spain would become bitterly anti-Italian, and thus pro-Allied. Spain would stay out of the war if Italy remained neutral, and also after France was crushed, but unlike OTL would not cooperate with German intelligence, or provide the volunteer Division Azul for the Eastern Front.
The Balearic Islands would allow Italy to dominate the western Mediterranean. Any Allied ships going from Gibraltar to Malta would be under air attack starting over 500 km west of OTL. Aircraft ferry missions would be extremely risky.
This would make the North Africa campaign harder for the Allies.
Supposing the Allies decided on an Operation FLARE to gain control of French North Africa; they would not dare land inside the Mediterranean. Oran, the first target east of Spanish Morocco, is less than 400 km from the Balearics. The Allies would have to land in Morocco only and move west overland.
Once the Allies had Algeria, there would have to be an invasion of the Balearics before any other moves.
An alternate form would be if Italy did not annex the islands, but merely bullied Spain into allowing an Italian base there. This has military and diplomatic complications.
In the post-war world, having forces of one country based in the territory of another has been commonplace. In 1940 it was unusual. If the basing power went to war, there might be battles between them and their enemies in the host country's territories, with the host country not involved. Aside from Egypt I can't think of any WW II examples.
If the Allies invade the Balearics to take the Italian bases, that might lead to war with Spain. It might be only a
de jure war, with the Spanish refusing to take sides.