I would really like to see this problem addressed where it should be addressed—at the browser level. At the moment, a MITM attacker can intercept your non-encrypted requests, mess with them, and trick your browser into sending requests with arbitrary content to the sites that you care about. It's this interaction that's making several very interesting attacks possible: BEAST, CRIME, RC4, and now BREACH.
A carefully designed opt-in security measure could do the trick, but I suppose it would take a lot of talking and politics to get it implemented. The idea is that a web site can control which other web sites (cross-origins) can initiate requests to it, even if it is via script and img tags.
Incidentally, just a couple of days ago, Mike Shema and Vaagn Toukharian (fellow Qualys employees), proposed a new cookie control (there is now a follow-up post from Mike) that would restrict when cookies are sent. Their intention was to deal with CSRF, but the measure would work against BREACH, too. If a 3rd party web site initiates a request to your web site, against your wishes, being able to tell the browser to drop the cookies would mitigate the potential attacks.
Update: The first version of this article included a referrer check defence that allowed empty referrers, with the idea to support auto-log in for the users following bookmarks. But then Krzysztof Kotowicz pointed out that the Referer header can be removed by the attackers. I have now modified the example to drop cookies on all requests not originating from the web site.
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