If you want that your user be logged in both site accesses, have a look at : http://ez.no/doc/ez_publish/technical_manual/3_10/reference/configuration_files/site_ini/session/sessionnamepersiteaccess
I think that you also need the custom sessionnamehandler : http://ez.no/doc/ez_publish/technical_manual/3_10/reference/configuration_files/site_ini/session/sessionnamehandler
Thanks for the links. What you pointed to is not was I was looking for, but that's some very interesting functionality that I had not known about before, so big THANKS!
I've just solved my problem, and ashamed to mention that...