Did you CHECK "Users must be registered and logged in to comment"? (Settings > Discussion)
Then UNCHECK "Anyone can register" (Settings > General).
If that doesn't do the trick there are plugins that can help. Here's one. Be sure to read all the plugin documentation before doing anything so you are clear about what to do first and how to proceed.
If that doesn't work with your theme, then some PHP snippets can be added but you'll have to create a child theme first because we don't want to modify core WordPress files.
That didn't work, and the plugin described doesn't seem like it would alter the theme. What files would I need to copy over to the child theme before making these changes/adding php?