Funny enough, by posting the above steps I may have discovered the actual repro steps (or perhaps just another way it happens, not really sure). As I was writing up the previous post, the bottom of the page correctly showed 1 unread. However, as soon as I submitted that post it changed to 2 unread, which was wrong. Clicking the 2 unread link brought me to /unread where I verified that there really was only one unread topic. It’s as if submitting the post added to the unread count even though it was my post and shouldn’t have affected the unread count. If this happens consistently it should be more useful to reproduce the bug than my previous steps.
EDIT: Lo and behold, when I submitted this post it did change incorrectly to 2 unread again. It stayed that way for several seconds before changing back to the correct 1 unread. So I think these steps should work. 
EDIT 2: I posted in a topic I was tracking in https://community.wanikani.com, and in this case it also went from not showing the unread link to incorrectly showing 1 unread. This time it took a full minute for the page to realize I didn’t actually have any unread topics at which point the link disappeared again. I’m now three for three with the steps in this post reproducing the issue, so I think these steps will reproduce consistently.