Heard this theory a few times, sounds like the kind of thing Churchill would do but never seen any proof. Basically according to the theory he was worried about communist fighters in Mediterranean countries and their post war influence, and wanted to make sure as many as possible were dead.
Thanks in advance.
No, this is not correct.
First of all Churchill was consistently calling for the outright full-scale invasion of the "Soft Under-Belly" of the Axis the entire war. Against the wishes of the United States and Russia he pushed for the invasion of mainland Italy, and then escalated the invasion once it was bogged down by personally designing and advocating for the disastrous Anzio Beach landings. These landings very nearly destroyed any chance of launch the invasion of Normandy a few months later (because of the lack of amphibious assault vehicles that were constantly needed to resupply the beachhead).
Churchill had early on also called for an invasion of Italy in the Adriatic as far north as the area around Venice.
There is no question that he believed the capture of Northern Italy was a strategic imperative, even though it turned into a strategic disaster.
The reason why the Allied effort to take north Italy was so slow is because: the Northern European Campaign was more important, the Germans had highly effective defensive lines North of Rome, and supplying the Italian front from the Sea was incredibly difficult, the Germans having destroyed all the Ports, and the new Italian "Allies" being complete reliant on war materials for their own people to survive.
Even under all of these conditions, the OSS and British paramilitary forces continued to drop supplies and personnel deep into "occupied" Italy in hopes of forming a coherent resistance behind enemy lines. The problem was that Italian "communist partisans" were few, having lived under fascism for decades, and they also were not very effective. They had little or no military training and largely survived by turning to banditry, which they inflicted on the civilians in the then Salo Republic.
See:
Also, a great book is the "Brenner Assignment."
Douglas Porch's "The Path to Victory: The Mediterranean Theater in WWII," is the definitive book on the this particular area.