This error has been driving me nuts. We have a server running Apache and Tomcat, serving multiple different sites. Normally the server runs fine, but sometimes an error happ
8 updates of the question later one more issue to use to test/reproduce, albeit it might be difficult (or expensive) for public sites.
You could enable https on the sites. This would at least wipe out any other proxies caches along the way. It'd be bad to see that there are some forgotten loadbalancers or company caches on the way that interfere with your traffic.
For public sites this would imply trusted certificates on the keys, so some money will be involved. For testing self-signed keys might suffice. Also, check that there's no transparent proxy involved that decrypts and reencrypts the traffic. (they are easily detectable, as they can't use the same certificate/key as the original server)