It is pretty close to having real time auto-updates (you do have to click the notification or refresh the page to see the tweets though). As Twitter said:
We’re starting a limited test of notifications on twitter.com for when you have new tweets. So if one of the folks you follow has tweeted since you loaded your homepage, you’ll get a little notice saying “1 new tweet” that, when clicked, will display the new content.I agree that this is a very nice improvement. I will be even happier when the notifications are applied to lists which I follow. That is the main way I keep up with my stream now, and hope to see the two new features integrated soon.
We think it improves the tweet consumption experience and hope to roll it out to everyone as soon as we can.
UPDATE: WOW! The new feature shows in my listream as well. This is really making Twitter much easier to use... I have three different tabs open with a separate list in each and get notifications when I have new tweets for each one :-) This now makes my browser the best client, which might be by Twitter's design. But I am sure that as third party apps integrate lists, notifications and retweet, then I will go back to using my old favorites.