Discovered that there seems to be a bug in Discourse’s tag management.
We have a small number of tags grouped into a Third Party Addons grouping.
Even though the preferences are set to all the tags to NOT be restricted, those tags are still not available globally. (Not only were they missing from the menu, the ‘create’ feature was disabled if the term was entered manually.)
I noticed this most frequently with the supertool-script tag. (Which lead to the creation of a supertool_script tag, causing further problems.)
Removing the tag from the grouping has made it global. The supertool_script tag was also set to be a ‘synonym’ of supertool-script … so that problem goes away too.