I got a weird problem involving both of my cats (Siegfrieda, to the left; Kika, to the right).
Kika is rather particular about having her own litterbox(es), and refuses to use a litterbox shared by another cat. Frieda on the other hand is adept to the "if I fits, I sits, I shits" philosophy, and is totally OK sharing litterboxes.
That creates a problem: no matter if properly and regularly cleaned, the only one using litterboxes here is Frieda. We had, like, five of them at once; and Kika would still rather do her business on the patio.
How do I either teach Kika "it's fine to share a litterbox", or teach Siegfrieda "that's Kika's litterbox, leave it alone"?
I'll let you know if I find a solution. Your situation reminds me my sister's cat though - he "used" the litterbox halfway (his front paws inside, his back paws outside), and then did it on the floor next to the litterbox. Might be worth checking if that isn't what's happening. (She solved it with taller boxes)
I've tried several different boxes and litters. We have five boxes but they all 3 want to use one. The problem kitty was my mom's. She would use the box, meow, and my mom would scoop it immediately. Instead of going to use one of the other four boxes that are completely clean, she poops on the floor beside everyone's favorite box.