Well, poop
Finally called the Orkin man, and he took a close look at the ant situation. Turns out the ants are termites.
I wish I had figured this out earlier. There are plenty of information pages about ants and termites online, and one look at the pages could have told me instantly that these critters are termites.
When I first moved here, people told me that termites were nearly nonexistent in this part of the country. This false notion led me to ignore the possibility. They looked like ants, so I assumed they were ants.