How to get the post editor and media library to respect HTTPS administration mode?

I have a site with FORCE_SSL_ADMIN turned on. When I’m editing a post with an embedded image, or when I’m viewing images through the media library, they are loaded using the http:// protocol rather than https. This causes a mixed-content warning in IE, which clients kind of freak out about.

Does anyone have a recommended approach for finding/replacing these non-secure image URLs while viewing them through the admin?

The frontend loads over regular http, so I don’t want to do any replacing in the DB that will force images to load over https outside of the admin.

I figure this has to be a common problem, so I’d appreciate any advice, even if it’s “you can’t do that”.

You can’t right now. This needs to be fixed in wordpress core. Probably this can be temporarily circumvented for a fraction of the problems with a plugin, but it’s much more valuable to invest the time developing something in an actual fix of wordpress.