About the only control I’ve found for these tough, invasive grass/weeds in clay soil is wait for a nice early spring day when the soil is moist and deep till thoroughly, then rake and pull out all the rhizomes you find and let them bake dead somewhere, not in the compost pile. Ideally, you’d repeat the process because there are more under there. Add as much organic matter as you can to the soil at the same time, to keep the clay soil loose. Then, as the season goes on, grass sprouts that do appear pull up fairly readily along with their roots.
At the same time, I use my Mantis to till a “dead zone” around the garden beds to keep the grass from moving in from ouside. I run the tiller through every few weeks, and it seems to work. Of course, both these virulent weeds spread by seed as well, so your work is never done. But you can pretty much break the back of the beast.
If y’all have other solutions I’d love to hear them.