Please consider opening at Tableau Support ticket for this at https://support.tableau.com so that one of our support engineers can engage with you on this to see if there are any other options.
Thank you Patrick! Good news. Since the server upgraded to v2018.01 I'm no longer able to replicate the issue, the browser is working as expected and prompting for login in any browser window when going to a new link.
Great to hear Daniel. Thank you for coming back and marking the answer correct and sharing the action that resulted in resolution.