Redmine

I've set up 2 redmine instances at different URLs on the same server:

project1/redmine
project2/redmine

Unfortunately it looks like redmine does deal with this well as it only stores 1 cookie and so switching between browsing the 2 databases causes me to re-authenticate. Is there a way to get this to work with out having to have different hostnames for the different projects or would this be an easy change to make to redmine so that the authentication info for both sites was kept in the same cookie?

Cheers

Russell