[jmsgwd]

OK but now I'm puzzled why some of my encrypted backups have this problem and some don't. I thought I followed the same process to create all of them.

Here's what I did: I used Disk Utility to erase and format the drive as "Mac OS Extended (Journaled, Encrypted)", and set an encryption passphrase. The passphrase is random and different for each drive (which is generated by and stored separately in my credential vault application).

Then I used SuperDuper to (initially) Erase then Copy; and used Smart Update the drive. The SuperDuper Erase always seems to preserve the encryption (along with passphrase) that I previously set up in Disk Utility.

When I reboot the drives that work, I am first prompted for the FileVault passphrase, and following this, it boots into the normal login screen with all the users there.

But maybe I'm remembering wrongly, and I in fact followed some different steps to create the encrypted drives that boot correctly.

Another change I've made recently is that I've now encrypted my main system drive as well. I understand that this changes the way OS X boots up, so I now get a pre-boot login screen rather than a post-boot login screen, and it uses the passwords of the OS X users to decrypt the drive before completing the boot process. But I'm sure that post-encrypting my system drive, I've followed the above steps to create a bootable backup and it's still worked as described.

Also in your reply you said "reinstall the OS to recreate a recovery volume on the backup". I don't understand what this means. If I re-install the OS won't that overwrite the backup?

Maybe what I need is step-by-step guide to creating a FileVault-encrypted backup the right way, assuming the original system drive is also FileVault-encrypted, which explains the whole process from beginning to end.