Thanks, DrByte, this is driving me mad! I posted the site's Configuration->Sessions and the PHP session configuration in the first post of this thread. The only thing that's changed from that posting is that I now set the session.use_only_cookies to 0.

The SSL cert's been active for "a while" (not recently changed) and the hosting has been constant for the past year.

I don't tend to use IE11 (unless I have to) or Chrome (except to debug stuff), so there shouldn't be any stale cookies lying about, but I'll reset the cookies on both browsers just to be sure.

It's weird, because there's no stickiness to the zenid when the site's accessed via FireFox.