I managed to get the tagging system working again.

It appears that a WordPress plugin I’d been using to manage tags and add tags to posts was either screwing with my existing tags or is simply not compatible with WordPress 2.6 (to which I recently upgraded.)

The plugin in question was Simple Tagging by Amaury Balmer. But bear in mind that it’s also equally possible that I, as a well-documented idiot, screwed something up. However, I’ve deactivated the plugin and have started using a different tagging plugin. Things appear to be OK.