Apparently, Zeos had the same arrangement on their Pentium machines, and it seems I do need power supply with a third plug supplying 3.3V, or some kind of adapter.

You can use ATX power supply with AT adapter and make additional 3.3V connector (orange wires).

Ok, so I figured what's going on. A small number of late AT power supplies included a third motherboard plug, named P10 or AUX. It does supply 3.3 volts, and was a thing on a few OEM systems from Compaq, Zeos, Unisys, and probably various other. I could not find such a power supply on Ebay, and there's hardly any mentions even on Usenet -- this must have been rare, even back in the day.

I found a site which supposedly still has an ATX to P8, P9, and P10 adapter in stock, and I also bought a 200 watt Seasonic power supply with the old -5V line since I don't have any other power supplies, ATX or otherwise.

I have one of those P8, P9 and P10 adapters, basically the P10 takes 3 x 3.3V and 3 x grounds from the ATX power connector. I have a couple of boards with that type of connector, an Intel Batman's Revenge (Socket 4) and a Zeos Pantera. In the Intel board for example, that connector is there only to support 3.3V PCI cards, so it usually isn't necessary since most PCI cards from that era are 5V. The manual of the Zeos board on the other hand claims that the P10 needs to be plugged in when setting the 3.3V jumper for the CPU voltage.

One problem with the adapter I bought was that the cables were very short, which would cause problems if the P10 is located far from the P8-P9 on the board... so you may end up needing an "adapter for the adapter" in that case.