I think I've got it. (But the cuttings in plain water never showed any
roots. That just didn't work.)

As I said above, aquarium with plexiglass on top. Put sterile potting
soil in six inch tall foam cups slotted in the bottom. Get a cutting
with that length of stem (I'm just doing Red Cherry), and pack it in
to the soil, with the tip of the stem on the bottom of the cup. That
way, you have six inches of stem in the soil. Soak the cups once with
sterile water. Put cup in aquarium (with a half inch of water on the
bottom), and mist once every day or two. Kept indoors, next to a
window with partial sun. Temps stay 76-80F inside, and aquarium walls
are covered in condensation continually. Relative humidity in there
must be ~100% all the time. So the key ingredients are soil and
humidity, it would appear.

Started that about two weeks ago, and not only are the cuttings
surviving, they're growing! Never even any wilting. Some are now
pushing up on the plexiglass lid.

Hooray!

I wonder if the variety of tomato makes any difference here.