I opened the relay to see where the coil is and how the magnetic circuit is shaped. I also wanted to see where the permanent magnet is. It should be somewhere, because how else did I get the signal far away from the magnets of the original pickups?
I've read that relays with permanent magnets are a thing, and maybe what I've got is one of them. I also thought about the Earth's magnetic field as the other source, but couldn't really believe, it is strong enough to get any audible signal.

I couldn't detect any magnetic field inside, so I just shrugged and ordered a toy oscilloscope to study how signal shape and amplitude changes depending on coil position.