Ok, I'm convinced. I hold off on tilling. One other thing I should have mentioned is that there is no way to get anything into the backyard without going through the house, so sod is going to be awkward. But, I never heard of a sod cutter before and it certainly makes sense that tilling clay can be messy.
Instead of heading off to rent a tiller, I'll look around at sod cutters and the availability of sod. I suppose they put roundup all over the sod but at least I won't know about it. Maybe.
Thanks for the advice. I'll head over to the local Garden World for a start.