I tweeted about it before: I think the Twitter website isn't realtime, because the little number in the title of the webpage makes people click the blue label that shows there are new tweets. It therefore increases the engagement on the Twitter website.
If I have Twitter open in my browser I will click the blue box to get a few pellets with new tweets. I can't help it. It's also the main reason not to use the Twitter website (apart from this I actually quite like it).