We have a dog bag. No idea what they are really called but it has a velcro collar and zip up back. Open it up, put now grumpy doggie in, zip it up and wait. Dog dries, mud collects in the fleece material and is sort of ok.
It worked well enough last week when we spent 4 days / 3 nights walking about the muddy woods South of Dumfries.
However, parked up at lunchtime today and I saw a dog walker with what looked like a 10 liter bottle with a hand pump to pressurise it with air and a spray nozzles being used to wash doggie...
So there are many solutions other than AstroTurf...
Cheers,
-G