That is indeed the California coupe. It appears that a previous owner added a soft-top and had a custom zippered enclosure made for it. This is why there is "no room to even have the seatback in it's upright position."
If you want to convert this to the conventional setup (I'm not suggesting whether or not you should), you'll need the soft-top compartment, aluminum lid, soft-top/trunk hinges, release cable, release lever, etc.