My dogs are tracking way too much mud into the house.

The problem is, I'm in a rental, and the 'lawn' is really just mowed grass and weeds. The soil is extremely well drained glacial gravel, with no measurable clay or organic matter to bind the soil together. Between the dog traffic and the gravelly soil, I've got a lot of bare spots, particularly right at the base of the back steps - hence the muddy paw prints all over the house.

Now I know exactly how to install and maintain a good quality lawn, did it professionally for years, but I ain't gonna do it here - won't be here long enough and no the landlady isn't going to pay for any renovations.

What I'm looking for is suggestions for something to cover up the bare spots that will stay in place when two wild dogs go barreling in or out, chasing whatever they thought they heard out there (I have an open dog door policy). I have lots and lots of fallen cherry leaves right now and I was kinda thinking of layering them on the bare spots, but they wouldn't stay in place very well. I wouldn't mind leaves being tracked in nearly as much as the mud.

Is there some kind of erosion control material I could lay on top of the leaves on the bare spots that wouldn't trip up the dogs running over it? Or would the leaves stay somewhat in place once they were wet enough? we're not talking a huge area here. I don't want gravel or stone because it would be kicked all over the place. Bark and sawdust would be as messy as the mud. Tree chips seem to be uncomfortable for dogs to walk on and also get kicked all over. What about landscape fabric?