Server 500 in CMS after logging out of the Console

Last week nvankampenhout already reported a similar problem, HIPCON-2287, for which I now have a reproduceable case:

1. login on the .org CMS and open (just view) some document
2. login on the .org Console and logout again
3. back in the CMS navigate around the documents/folders, which still works
4. now try to switch perspective, e.g. click on the Dashboard: no effect
5. hit F5 (refresh): HTTP Status 500 -

The only way to 'fix' the above (underlying problem: session lost?) is to either open and login (and logout) the CMS in an anonymous browser tab.
Or (easier, but even scarier) login to the Console again in a separate browser tab and therafter you can simply refresh again the CMS tab, which then turns out to be still connected/logged in...