Your cat might have a bladder or kidney stone which is causing extreme pain when pooping. Please take your cat to the vet and have that checked out.
I ran into the same thing: Cat pooped in exotic, awkward places, but never the litter box. Never looked like she was in pain -- just being stubborn, I figured. Turned out to be a bladder stone brought on by the ingredients in modern cat food (put in place to help other feline health issues, but which causes stones in a small subset of cats). After the stones were cleared up, all litter box issues disappeared.
..but they do love some simple pebbles, which are, fortunately, dirt-cheap (or pebble-cheap :) ) although a little messier to clean than the above alternatives. I think they'd like sand as well (I live in Uruguay and those pebbles are from Argentina, so I don't know what alternatives are in the U.S.).
"Yeah that's great and all but I've tried that. 3 litter boxes. All cleaned twice a day, and he still shits on the carpet 2 feet away from the box. Lid on, lid off, big sidewalls, different cat litter. I've tried it all. The little shit just likes shitting on carpet I guess. And putting the fear of god into him does nothing."
I ran into the same thing: Cat pooped in exotic, awkward places, but never the litter box. Never looked like she was in pain -- just being stubborn, I figured. Turned out to be a bladder stone brought on by the ingredients in modern cat food (put in place to help other feline health issues, but which causes stones in a small subset of cats). After the stones were cleared up, all litter box issues disappeared.