Hey Aaron --
The session established when your user authenticated with Trusted Tickets is still open. I suspect that if you have the user log off (which they may not be able to do since you're probably not showing Tableau Chrome), your scenario will work fine. FYI, one currently can't log a user off programatically, but the feature has been requested for the JS API....
The reason it works when you close the browser is that the "trusted ticket session" isn't re-used.