This cat of yours reminds me of one of my Mom’s cats who would always do that when his litter box needed cleaning. Perhaps in a previous home she got used to the box always being in need of cleaning. I’d try shovelling the poop into the box, put her in, and praise her while she’s in the box. When she goes back to the box and finds it there later, she might get the idea that’s where it’s supposed to be. I would also keep her box in an area where I could keep an eye on what she’s doing and reward her when I catch her doing it right, and try to correct her before she gets it wrong. Most cats like to dig a bit in litter or soil (big plant pots are favorites), and I think it’s their instinct to burry their poop or something. You might try encouraging her to dig in clay litter – try the clumping kind that’s finer and maybe easier on paws. Perhaps she misunderstood some sort of punishments in a previous home, and learned the wrong thing. It’s also possible she may have been an outdoor cat who never used a litter box much if at all – if so, that lack of human contact may also explain some of her shyness? If she’s sitting in the litter box, but the poop keeps landing outside of it, try a covered litter box where it can’t.

Is she just ppoping outside of the litter box, or does she urinate outside of it also? I had a friend who had a cat that did that, and all it turned out to be was a urinary tract infection. also, some cats are real finicky about how clean the litter box is…clean it everyday if that is the case. Good luck…keep us all posted.