Production caching causing languages to get stuck

We have a site using the up-to-date plugin at http://latinameryc.wpengine.com/

Our issue is that if you click around to various pages, switch to Spanish, then switch back, eventually this behavior will cause the language to get stuck in Spanish. It happens on any page (parents, children, blog, custom templates, etc.) Below are some notes and troubleshooting steps we’ve performed:

1. several developers and various people have been able to reproduce the issue in various browsers, devices, OSes, and networks
2. the issue cannot be reproduced locally, on any machine or OS
3. the issue cannot be reproduced on the host’s staging site
4. the issue only occurs on the “live” site, which uses nginx, apache, varnish, and EverCache (WP Engine’s custom caching tool)
5. we have debugged the issue on the server-side with WP Engine, and can isolate the issue to their production environment (with the stack detailed above), but they’ve asked if there are any cache or cookie files that they can exclude from caching, since the issue can only be reproduced in that environment