Earlier quoted context omitted.
Gmail allows you to sign in to multiple accounts at the same time (in different tabs). I have two pinned tabs always open in Chrome - one for personal, one for business emails. Works incredibly well for me.
How does this work? If I sign out in one tab, it signs me out in another tab, on Google Chrome. Just tested it.
Disclaimer: I am about 99% certain this is not some weird internal Google feature (not on the gmail team, don't worry), but it's also been > 1 week since I had to go through this performance so I may have forgotten some detail.