I've struck out with even a guess where to start here.
http://co-op.downinthecountry.com/wiki/
No browser (Firefox, Safari, Chrome, IE) on any of 3 machines at this location can remain logged in. Move away from the page on which the login was performed, and you're logged out.
However, others not at this location are able to retain logins.
I shudder to think that it's a HughesNet issue (the very best of clueless Indian support), but that's the only thing that's unique to us and common to the 3 machines here.
The HughesNet rig is an HN7000S into a NetGear Ethernet Switch.
I have no clue about where to start troubleshooting.
Thanks for any thoughts.