ADDED: In general a lot of services don't handle multiple accounts terribly well. I've had issues with Wordpress as well. I actually have two different soft-key authenticator programs related to making personal and work accounts play nice.
Sounds like you're using the Google Accounts feature, and using the [separate] browser profiles feature would exactly solve your problem.
In particular: you can log into different profiles in the browser (each window can be associated with up to one profile, and they can be different), by clicking the little "Sign In" tab in the upper right corner. See also https://support.google.com/chrome/answer/2364824?hl=en&co=GE....
You can then open tabs under the different profiles with right-click, "Open link as", and selecting the appropriate profile.
(Personally, once I tried profiles, I never touched multiple-accounts-logged-in again.)
ADDED: In general a lot of services don't handle multiple accounts terribly well. I've had issues with Wordpress as well. I actually have two different soft-key authenticator programs related to making personal and work accounts play nice.