[In Omnibus]

The cookie prefix is likely the issue. The cookies for both sites would have to be in the exact same location with the exact same prefix so that both sites would read the same member login information.

Match your cookie domain setting for both URLs and then change the settings in:
Settings/Options/Cookies And HTTP Header Options/Cookie Domain

This option sets the domain on which the cookie is active. The most common reason to change this setting is that you have two different urls to your forum, i.e. example.com and forums.example.com. To allow users to stay logged into the forum if they visit via either url, you would set this to .example.com

So, it appears as though the cookie can only be active on one domain unless both URL domains are sharing the same cookie. Set both domains to read the same cookie and your issue will be resolved.