They have different transmission probabilities. Omicron has far more presence in the upper respiratory tract. It's just a numbers game, they're not competing in any active sense.
In that case I don't understand the "omicron will outcompete other variants" thing I keep reading about. If they don't interact wouldn't the progress of one variant be unaffected by the presence of another?
I think you're correct - if Omicron and Delta provided no cross-immunity, there would be no competition, and they would just continue on their own trajectories as if the other didn't exist.
I'm not an expert (so take this next part with a very large grain of salt), but I would guess that someone with an active Omicron infection (and so swarming with antibodies) would get the biggest immunity boost against Delta. So if Omicron is everywhere, Delta runs out of infectable hosts. In principle, it could even go extinct - if that were to happen, it wouldn't matter so much if Omicron-induced immunity wanes, there would be no Delta left once the shields are down.
