When we set up the Aphlict server, I expected such an awesome experience from Phabricator as well. I pinned Phabricator in my browser, turned off all emails and started to wait for the tab to highlight to alert me of a new diff. Sadly, it didn't highlight, and I didn't notice the new diff.
The implementation to get the tab to highlight should be relatively simple. All that is needed is a change in the tab's title. Both GMail and Twitter do this by inserting the number of new messages in the title element. Asana does it very nice by only adding a icon in front of the normal title.