No animals at all? I am in an urban environment with a 19'x35' postage stamp of a backyard, and according to the bylaws, I can still keep fish and
rabbits, along with the usual dogs and cats. Your situation might be harsher, but I have been thinking that I could either have rabbit tractors, or just keep a herd in a giant hutch built into a hugelbeet and drop the cuttings in with all my kitchen veggie scraps.
This year, I trained my morning glories up my tomato vine supports and pruned them of all but a few leaves out of the way of the tomato plants, and I have the morning glories binding the tomato vines to the supports. This works for me because my tomatoes get some brief but regular individual attention, and so it's not much more work to keep the morning glories under control.
If you are okay with manual weeding, just chop and drop. You won't believe the improvement to your soil. Morning glories are hyperaccumulators, right? Maybe control as opposed to elimination is a good idea. Also, with your clay soil (with which I can sympathize), it might be a good idea to trim individual plants at soil level, leaving the
root structures in place. They might send up shoots that you will need to control in turn, but living or dead, the root systems are allowing for better air and
water penetration to the root zones of the plants that share space with the weeds. And if the weeds' root systems do rot and die, then you've just created a root mass-shaped network of
compost.
-CK
A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.
-Robert A. Heinlein