4 way switch circuit troubleshooting
To me it is a single mis wired 3 way switch, @jack was almost correct (all) 4 way switches swap the 2 travelers from 1 side to the other+.
Replacing a switch is allowed if the landlord approves in all the locations I have worked as a licensed electrician.  Some jurisdictions like NY a home owner can’t do basic maintenance so it is best to know.
I would check to see where the hot is coming in on the switch note the connections photograph them or record the connections if you don’t have a meter.
It sounds like a traveler and common at one of the 3 ways is reversed.
The good thing is if you have access to the breaker panel (you should by code) this is easy to figure out trial and error with in 4 tries.
Pick 1 switch power off swaps trailer and common turn power in and test  did not fix power off now the other trailer is moved to common swap and test no better return to original and go to the other 3 way and repeat the same sequence.
There are 2 travelers at each end  one of these is mixed up with a common. If incorrectly connected it doesn't work properly in 1 or 2 of the combinations there is no damage possibility using the same wires on the 3 terminals (don’t touch the ground) all the 4 way dose is flip the traveler pairs.
It can be faster for an electrician to do this test than take measurements and make the change I have done this many times with the exact same issues and had success.