You kind of can / cannot be in more than one environment at the same time...
If you use chrome, log in to say your SB account always make sure you have at least one tab that is an SB page.
When you want to dip in to Prod navigate someplace like Home and switch to that environment.
Make sure you still have your other SB tabs to one side.
Once you're done in Prod switch back to SB in the account list and your old SB tabs will spring back in to life without having to log in to each in turn.
Once you get in to that pattern it's not too bad just make sure if you accidentally navigate to a tab that's not the environment you expected that you don't click the login button as it'll dump you in to it again. (although old tabs will still maintain the content you'll just have to switch env again).
Hope that makes sense, because I'm off for dinner now 😉