Our tape backups haven't been working properly for a few months. I've been really busy and we have nightly off-site backups anyway so I haven't taken a look at it until now.
Initially I thought the backup script had gone crazy and relabeled all the tapes with the same GUID: it complained about tape 1 being in the drive whereas it expected tape 2. Tape 2 was in fact in the drive. Same happened with our other tapes.
After poking around ntbackup and the RSM I noticed that the tape drive had a big red X next to it under the "Libraries" container in Computer Management -> Storage -> Removable Storage. That made me realise that the GUIDs hadn't actually been overwritten, just that the last tape's GUID (and that would be tape 1) had been cached in the RSM's database from before the red X appeared.
After inspecting the backup script's log I noticed the date it stopped working was close to the date I had removed the server's virtual machine snapshots (in a very roundabout way due to constrained disk space, I ended up with a new virtual machine). That made me realise that I hadn't added the tape drive to the new snapshot-free virtual machine's configuration. A couple of clicks after and the backups are working again, yay!