I've narrowed this down to the Global Site Tags plugin. When I update widget-global-site-tags.php the widgets are still there. When I update the post-indexer.php (Though it looks like it has the same date as my old one so it might not be an upgrade) the widget still shows the tags, and the tags page still works.
When I update to the most recent version of global site tags, I get:
"There are no tags to display."
on both the tags page and in the widget area.