Grass is Greener

Daikon radishes. They grow in about anything and are especially good at clay busting. Grow a bunch then let them die back. Till them in and repeat until you get enough environment for the worms to take over the tilling. You can keep piling on radishes with something like clover and peas to add some nitrogen fixers. This is more a pasture revitalization technique, but if you don't mind being the weird radish guy for two or three years (depending on local conditions), you could do it on a smaller scale for a lawn

So it is indeed greener where you water. Try the body thing next and let us know.