So it turns out that tiling a floor is a 3 or 4-day job. The work isn’t long during the “work” time, but you have to wait for stuff to dry before you do the next step, so at a minimum it’s 3 days, and 4 if you want to be careful.
The first step is the underlayment — called “backer board” — whatever sort you are using. You spread thin-set (another name for concrete) on the floor, then lay your underlayment, let it dry, then start laying tile on top of it.

I’m not entirely sure I should have taken on this project, but I’m doing it anyway.

Demolition begins. Yeah, that black stuff is the reason we’re doing this. It hadn’t destroyed the sub-floor, but had swelled it some.

Floor’s ripped out, thinset is mixed, making a mess, I’m cheerful.

That goop is the thin-set, with a 1/4×1/4 trowel atop it. I ended up wishing Home Despot hadn’t been sold out of margin trowels by the time I was done. And that orange stuff is a plastic type of backer-board that I’m trying out. It’s very light and convenient.
Tiling
DITRAMAT!!!! I was actually clicking on this when I saw you mention backer board to post what Ditramat is and how WONDERFUL the stuff is! Tiled the entire great room of my old house 2 years before we sold it, and never regretted it!
So much lighter then backer board, thinner then backer board, and you can begin setting tile on it IMMEDIATELY!
Chris