How can Western Allied Troops end up fighting alongside the Soviets on the Eastern Front

In OTL Stalin was strongly opposed to allowing troops and aircraft from the Western Allies being stationed on Soviet soil and fighting the Germans on the Eastern Front. He obviously accepted massive quantities of foreign aid (While generally doing everything possible to prevent Soviet Citizens from realizing how much the Allies were helping them. For example they went through the trouble of removing labeling from foodstuffs that would indicate Western government origin. For instance canned food from the US was relabeled as being aid from non existent American communist groups.) but opposed having Western troops operate from Soviet soil. The US/Britain wanted to use Soviet air bases for raids on Germany. Stalin opposed it and when eventually he sort of relented did everything in his power to strangle the effort.

So how bad shape would the Soviets have to be in to request signifigant Western Allied ground and air formations fight alongside Soviet troops on the Eastern Front?
 
A successful German drive into the north Caucasus, isolating Trans-Caucasus from Russia, with a further threat of Axis capture of the Baku-area oil fields. This might force Stalin to accept US/UK troops from the Middle East in that theater.
 
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