On an empty tag wiki page (e.g. here) it says that
Tag wikis can be edited by users with more than 1500 reputation, provided: They are in the top 20 answerers for this tag or they hold the bronze tag badge for this tag.
To edit this wiki, you must have at least 2000 reputation and hold the bronze tag badge for this tag.
Further, the privileges page says that one needs 1500 rep and
either be in the top 20 answerers for the tag, or have a total score of 100 or more for the tag in question.
The last clause is not quite equivalent to a bronze tag badge (which also says 'in at least 20 non-CW questions').
This is clearly inconsistent.