When you say 'everything is stable' do you mean that you've checked the water stats (ammonia/nitrite/nitrate) and they're fine?
Is there any tail fin left at all? If there's some fin left, if you treat the cause the fin will slowly regrow. If the damage has taken some of the flesh from where the fin grows out, there's no hope for him. If he can't swim without his tail (lying on his side sounds very uncomfortable) I would euthanise him personally.
If there is some fin left, what do the edges look like? Black, white, red or ragged edges may be finrot. This can be bacterial or fungal, either way it might be treatable. Are there any signs of white patches on the body or mouth, a slimy coating on the body, or any other spots, patches etc? Red or inflamed looking gills, mouth or skin? Any damage at all on the other fins?
Cories and algae eaters are peaceful fish and if they've lived together for months there's no reason why they should suddenly turn on their tankmate. Is there anything sharp he could have torn it on or any way that a bird (or an inquisitive cat or dog) could have gotten at the tank?