This computer has previously been synced with an iPhone or another IOS device

iTunes has stopped syncing my iPhone and iPad. Now when I plug either of them in I get the message in the subject line.

As a result it offers me the change to set up as a new iPhone or restore from a backup.

The closest I can get to explaining how it's got itself confused is that I had some kind of network issue on the computer, and I got a warning message on my iMac telling me that there was another device on the network called iMac, so it had renamed itself iMac (2). (There was no either iMac - I assume that on reboot somehow the network de-registration hadn't worked properly). I changed the name back to iMac and rebooted and everything else seems fine (including online backups that are done against the named computer). Not sure if this has anything to do with my sync problem or not - but it's the only oddity I can think of.

Any suggestions about how to get iTunes to allow the sync again with the iPhone and iPad that it was quite happily syncing with until recently?