I returned the original relay but found the same thing.
What I found was another with both internal coils at 24VAC in an "Ice Cube" package.
Evidently the original relay and all the others like it may have been removed from this kind of package.
The new one is an 8 pin octal socket which makes mounting easier.
It was getting late and I didn't notice it's only a SPDT.
A DPDT version to do both the AC line and B+ switching would have been better, but B+ switching can be shifted to the small relay.
The small SPDT (12VDC) relay will be for switching the series bleeder to ground from the 250V B+.
One set of 2 NPN's will alternately toggle the 2 large relay tails to common ground.
It turns out that they are DC coils after all, with a diode in series to each coil from the 24VAC input.
I can add a small electrolytic to each coil to slow things down.
I made sure the big relay worked this way before leaving the surplus store.
Depending on which relay coil tail is momentarily brought to common, the relay latches into that position.
There's no need to keep the coil energized as it is now latched.
A simple 7414 (or 4093) Schmitt trigger to debounce the reed switch to a 4013 (or 7474) FF to an NPN to toggle and hold the small DC relay with a capacitor, and to the other 2 NPN's to give a kick, 250ms, the big relay coils.
Sounds simple.
Wait a minute, do I need another relay?
There is no simple substitute to this mechanical issue given the constraints of where the switch is mounted and the mechanism rod returning to home after a press.
Has nobody tried to guess what this is all for yet?