Add extra parameters after permalink?

How can I add extra parameters after a permalink, specifically if I’m using a custom post type?

For example, let’s say http://mysite/album/record-name was the permalink. How can I make http://mysite/album/record-name/related not turn up a 404 or redirect?

WordPress doesn’t seem to call up the post template if the post doesn’t exist… so I’m at a bit of a loss how to do this.