Yep, same problem here, but luckily, no doggie door or my house would be full of mud

We did put gravel on one side of our fence, and put trees in (the trees for privacy & the gravel to get ride of the muddy path). Our neighbors on that side of the fence have 2 schnausers, and they too love to play tag! The gravel works, they can still walk on it & it keeps them from getting nasty. I'd love to have it around every part of my fence, but we have a large yard, so lots of hard work!
Maybe one of these days!

And I'm with the rest of you, when the carpet comes up, I'm getting a tiled floor in the kitchen and wood laminate in the living room, the carpet just gets too yucky with 3 dogs!