One theory that was discussed was the possibility that the 1 and 3 pop bumper EOS switches are either not opening or are shorted.
Rocket III Thumper Bumpers (resized).jpg
The 1 and 3 pop bumpers always fire together. If neither EOS switch opened at the bottom of the pop bumper stroke that would leave the two high power coils (~4 ohms each) each drawing over 10 amps for longer than the fraction of a second that they're supposed to.

That kind of load (20+ amps) on the 50 volt transformer output could put a strain on the transformer which would drop its output voltages. That might explain the dimming of the GI lights and the releasing of the Tilt relay which is usually active but operates at a lower voltage than the other relays and may be more sensitive to a voltage drop.

/Mark

Bally apparently used dual winding flipper solenoids in pop bumpers in games from the 1960s to 1970. The pop bumper drawing in the 1966 parts catalog shows a single winding (two terminal) solenoid but mentions that starting with '63 Hootenany that the dual winding E-184-190 solenoid was used instead. The drawings used in later catalogs all show a dual winding, or three terminal solenoid like the one below from the '68 catalog. Later catalogs point out that they switched back to a single winding solenoid starting with '70 Big Valley.
Bally Thumper Bumper (resized).jpg
The Rocket III parts list also shows that the pop bumpers and flippers used the same solenoid:
Rocket III Parts List (resized).jpg

I've created a new topic to continue the dual winding solenoid discussion to avoid further hijacking this topic:

https://pinside.com/pinball/forum/topic/bally-coils-with-dual-windings

Well done finding the short. I'll admit that having both EOS switches stuck on seemed like a long shot. But I think having all four pop bumper coils not only wrong, but designed for 25 instead of 50 volts is an even longer shot.

I'll echo @runbikeskilee's recommendaton to replace those coils. Over time they'll increase the wear on other parts of the game (playfield, pop bumper parts, plastics, ball, transformer, etc.).