Welcome to Just Answer. Are his stools normal? Is he doing this defecation inside where he sleeps / by where he sleeps?
The other possibility is that he fecally is incontinent and is having these stools just drop out without him being aware of it. If he wasn't moving his tail, then this would be a huge consideration. Cats are clean by nature and it should be his ultimate goal to not be sleeping in that small of an area with feces if he can at all avoid it. I'm curious if this is something that is happening while asleep.
I'm going to say this is just very, very strange to say the least. Is he treating this box as his outdoor litter box? For some reason, this may be the case. Could you make another box for him to use as a sleeping area and just let this one be the litter box?
Placing a regular litter box outside would also be a great idea. I hate to say this, this is interesting to say the least!