Ah,
I have never restored all data, only individual backups. I see now that the full backup backs up certificates and CAs.
When transferring my pfSense installation to a new machine, I have preferred to only restore some parts of the config (i.e., not the interface config due to network adapter differences) but I guess the easiest way will be to restore the full config, and change the network config manually from the console.
Sorry for the hassle.