So are you bridged then or is it really a double nat?

this is really where my limited knowledge of networking shows. I'm not entirely sure but I want to say both or double nat. There's two networks, but my router thinks theirs is a modem and is connected via the "Internet" port, not just a normal device port

Ah ok - sounds like a double NAT then, not a bridge.

Edit: to go into more detail, their router is acting as a NAT for your public IP, giving you your first subnet, and then your router is getting a single IP on that subnet and creating a NAT where your devices all get IPs. In a bridge there would only be 1 IP space behind a single NAT. In your case with a double NAT a lot of consumer things might not work (like UPnP) and port forwarding would require you to add rules to both routers.

Thanks, yes that's exactly what it is then. In order to give external access to my devices (e.g. my NAS) I have to forward ports from their router to my router, and from my router to the device. So, definitely double NAT. Amusingly the person who installed it incorrectly called it a bridge.

Thank you for the insight and lesson!