I now have a case where I can duplicate this every time. I'm certain it's not the exact same case as my former experiences (since I'm visiting a page I never visited before), but since it is easily repeatable, perhaps it will shine some light on the problem.
The steps are simple:
- Go to https://clerksupport.lds.org
- Sign in (note that the header reflects that you are indeed signed in)
- Go to https://clerksupport.lds.org
- Note that you are signed out, even though less than a minute has elapsed since you successfully signed in.
Using the clerksupport.lds.org URL does redirect to a different URL (one that starts with http://www.lds.org/callings/melchizedek-priesthood/), but I don't think that tells the whole story, since other similar URLs (such as leader.lds.org or clerk.lds.org) that also redirect to a different URL don't have the same problem.
The problem seems to occur on every browser I've tried, too.