B.C.
I knew someone who became a house flipper and he came across a house that had one outlet wired to another house behind the property.
He was working on the house and was sure he had ALL the power turned off but that one outlet still had power.
He called in an electrician who followed the wire - and at some point the guy said "Ok, this is off the clock. I just want to find where this thing is going" and it went across the back yard to another house.
The properties were 30+ years old.
The only thing they could think of was someone at some point had family living in the 2 houses and one was helping the other with the bills.
After properties were sold everyone forgot about it.
At any rate they cut and capped that wire and rewired that outlet properly.
I don't know how old your rented duplex is but sometimes strange things can happen with wiring.
I don't know how motivated your landlord is to finding/fixing the problem.
He should have an electrician go over all the wiring - and he might not want to spend the money to do that.
You might have to move.