The Italian "Theater" (1943-45) was in fact a continuation of the Axis defeat in North Africa, and was a Campaign for defence of Italy.
Indeed, this unification was realized after the Axis defeat of Yugoslavia and Greece in spring 1941.
El Alamein had not simply secured Egypt and the Middle East, but it contributed to what was, by any standards, a major Axis defeat in North Africa.
After the Axis defeat in Africa it was next involved in the Italian Campaign, and fought at the Battle of Monte Cassino.
With the Axis defeat in North Africa and the invasion of Sicily in 1943, however, the Italians signed an armistice with the Allies in September.
Following the Axis defeat in Africa, Allied forces invaded Italy and during a prolonged campaign fought their way north through Italy.
In winter 1943-44, season agriculture laborers became unemployed, while large transports of new internees arrived and the need for liquidation, in light of the expected Axis defeat, were large.
The Axis defeat on the Don river and during the Battle of Stalingrad however delayed their transfer.
As Egypt remained officially neutral until long after the Axis defeat at the Battle of el-Alamein, the Egyptian military did not participate in the Second World War.
Only about 40,000 returned to Hungary, scapegoated by Hitler for the catastrophic Axis defeat.