Boy, do I feel for you!

At my home, a small farm, when it is really bad mud season, I keep a bucket by the back door and dip each paw into the bucket and swish it around, before drying it off with a towel. Not ideal either ... but it works for me.

I don't have any idea what might grow in your area, we'll shortly have a good covering of snow which is much easier to deal with on paws than mud.

The only thing I can think of is landscape fabric ... but that's a mighty big yard and that could cost a lot.

I haven't tried to do that specifically, but I have sprayed down each individual paw (and myself, of course). I don't think a little dip would do it. It takes a direct spray to get the mud chunks off.

I would hate to confine them to just part of the yard when they love to run so much, but I may have to do something like that in self defense.

A bucket of water should work, I use that with my dogs if their feet get muddy or caked with snow on walks. You could keep a long handled brush in the bucket. I don't let my dogs run in the yard if it's wet or muddy as they could slip and hurt themselves.. but likely your dogs don't run as fast ;)

Contact your landlord and see if they'll either buy or allow you to buy some mulch. We just put some medium size Hemlock barkdust in DS's yard (two large dogs) and once it was in place and wetted down, their dogs could run thru it and not track in anything. The Hemlock is considered sliver-free so you'll probably want that for your dogs.

In my area a unit (7 yards) cost $230....money well spent.

I live on a farm to and I walk the dogs every morning. If they have run thru a muddy/clay area I make them run thru the grass. That doesn't always work so I make them lie outside on the patio stones until the clay has dried and them I take it off by hand. Most of the time when it has dried the dogs pick it off themselves. Rosie is the worst because she is an Australian shepherd with long hairs on her feet.

What about putting straw over the yard?

Dry pine needles work wonders for our dogs. Apparently, you can buy them as mulch also...