A clients site performs some magic to show different content per subdomain. By default each subdomain asks you to login, which is a bit annoying. It would be convenient if admins of a single WordPress installation could stay logged in when they jump between subdomains. Due to unique requirements, we’re doing this without WordPress Multisite.
I found that all we needed to do was set the cookie domain and path in wp-config.php. At first it seemed only COOKIE_DOMAIN and COOKIEPATH would be needed, but it did not behave until COOKIEHASH was also set. You could probably set it to anything you like, I just had a defined variable already.
* Set cookie properties to allow persistent login across subdomains.
define('COOKIEHASH', md5(MY_DOMAIN) );
This may also work between WordPress installations across subdomains. I haven’t tested this theory. Both installations would require the same login details and this config. It is working with WordPress versions 3.8+.