I don't think this is a Unified issue, since that is just a way of viewing accounts in a combined virtual folder. Do you have the account as POP on another device? That could be downloading from the Inbox and deleting from the server, the default POP setting, making it appear the messages have been moved from the IMAP Inbox.

The IMAP and POP accounts are on the same server and both reside on the Mac. The server name is either IMAP.sonic.net or POP.sonic.net. The IMAP account checks for new messages once a minute; on the POP account none of the five server settings boxes are checked. The POP account "waking up" would explain what I'm seeing since messages in the IMAP inbox that were tagged have no tag when they show up in the POP inbox.

If I tag an IMAP message and drag it to the POP account the tag is preserved.