Following.

I've used duct tape. Which is not the greatest option.

I've thought about putting a "safety area" of 1 row of puzzle mats around tatami mats but have no idea how and if that works...

This summer we get to try hard glue :D (glueing wooden stoppers onto wooden floor where no drilling is allowed).

In my old club I bought a wrestling cover that fit over our old tatami mats and locked around the corners. Worked like a charm and the old mats got a whole new life. Printed logo on it and everything. https://www.foeldeak.com/[...]covers.php

It doesn't, unfortunately. I had the same idea, and I tried this initially, but everything came apart and moved around, so I wound up just locking everything in place with a wooden border.

We put big bags of sand down on the edges to hold the mats in place with weight

Use an adhesive on the mats to keep them down and in place?

What kind of adhesive seems to work?

I'm not 100% sure on what it is called. I believe an adhesive that was used at my academy is the same adhesive used for carpets in commercial office buildings.

The way they laid the adhesive down was similar to the way cement is ls laid down for tile.

Do you want a permanent solution or a temporary one? When building a sprung subfloor I attached foam to OSB using contact cement which would definitely firmly (and permanently) affix mats to the floor. If you're looking for something where you can pick the mats up to make replacing them easier I think it's hard to beat a wooden border to hold everything in, but you could contact cement velcro to the mats and the floor so the velcro holds the mats in place (but you'd probably need a lot of velcro).