But there is definitely only ONE cookie?
If there are two cookies - one may be over-writing the other.
I had (for a few years) a setup like this:
http://www.bladekeep.com/forums/ - standard stuff
http://www.bladekeep.com/forums-lite/ - a VERY stripped down version of the forums for people to use at work (ie very few icons or fancy schmancy stuff).
I had no problem with this setup in terms of being logged in.