Eesh. Yeah, this one is really off the wall...
Best way I can see it is if, somehow, the Treat of Sevres manages to hold in most respects and the US joins the League of Nations (we're already talking a long shot and an even longer one). These can both be close equivalents. The Straits Zone becomes an international protectorate of sorts, but with the breakdown of Russia, more and more Entente troops, men, and material are based out of the straits to conduct operations in Crimea/Ukraine in order to support the Whites in Russia. This is also used as a base for American action in their mandate of Armenia (again, with the US joining the League) and Pontus (what the heck. It was too small OTL to even have a chance; that's why they merged with Armenia. But if we're going to go this far, let's have some fun) in order to help supply and expand the nation.
The French and British, their coffers and manpower depleted, eventually have to pull out. Their actions are somewhat successful, though, and a White state manages to be recovered in Kiev and most of the Baltic littoral. This, I see, is required, as the Soviet union might be enough to force the US out of Armenia de facto, with the Armenian republic being couped; a moderately strong Kiev government would distract them. Perhaps there is a Georgian republic/Azeri republic as well; that provides a cushion.
Of course, the American populace is very upset by all of this prolonged intervention overseas that seems to have no real purpose. There will be threats to pull out, as US troops and funds are being spent for no real gain. But the straits settlement, having gotten used to the fairly easy control (absent military zones) and dislike being turned into a warzone again. To satisfy all regards, the zone is eventually annexed and made an incorporated territory, providing an actual reason for the US to maintain operations in the region and keep it defended, as it is now US territory rather than international.
Now, I don't know how many loaded dice you'd need to make this work...