Even an antire male shouldn't mark in the house if they are properly house trained. They'd save it for outside.
We've had entire males before and they never cocked their leg in the house.

Maybe encourage him to go outside the same way you would just go to pee and give him regular opportunities. If you see him walking along something and sniffing in that way that dogs do before they are about to cock their leg, interrupt him and take him outdoors and encourage him to do it there instead.

Hopefully, you'll not have him at your house for too much longer.