search-backward returning matches which end after point

[Spencer Baugh <sbaugh@janestreet.com>]

search-backward (and re-search-backward, and looking-back) don't return
matches which end after point.  Is there a version which does?

For example,
(with-temp-buffer
  (insert "foo")
  (save-excursion (insert "bar"))
  (search-backward "foobar"))
 
fails to find the match.

Is there a function which will find this match?

Specifically, I noticed this when trying to use
easy-mmode-define-navigation to define navigation commands for a major
mode based on a regex.

The -prev commands defined by easy-mmode-define-navigation use
re-search-backward.  So if the point is inside the thing which the regex
matches, then thing-prev command will not go to the start of the thing.
But if point is after the thing, then thing-prev will go to the start of
the thing.  This is because of this behavior of search-backward.

I'd like to instead define movement commands which will always go to the
start of the thing when point is inside the thing.  But to do that for
an arbitrary regex/string, I need a version of search-backward which
will return matches which end after point.