Customers would call on the phone saying I'm looking at X, and it was kind of hard to figure out exactly where they were on the site by their verbal description. Where they'd come from (if they were referred from Google AdWords, they might have different expectations than from somewhere else). We'd have to awkwardly ask callers for the product ID number, or stuff like that.
To help with customer service, I built a real-time AJAX stream of all our Mixpanel events. (Which was easy since we were saving them anyway.) I found this to be essential for customer service and learning about our users in general. Sure it may seem creepy when you first learn about this, but honestly, what developer doesn't tail their server logs just for fun? With the right intention, it can be a seamless way to be extremely helpful to users. With a real-time stream, you're just suddenly more helpful to people who need it.
By anecdotal evidence, we have found that letting people know you were following them is not a good idea, b/c you may cause them to feel creeped out. But the truth is, it's more likely than not that this is already happening (to you, right now, on this very site). We would never deny it, but we tend to just not bring it up / volunteer the fact. A real-time stream just gives visibility into the data that's always been there.
What was great about having our own stream was that we can pull in any user account data that's helpful. In other words, once a user logs in, you can display any relevant user account info in the stream. How long has a user been shopping? How many products has he/she looked at? Is he/she a return user? Has he purchased before? You can essentially link your real-time stream with your customer relationship management, aka CRM. If you choose, you can reach out to them in real-time on the site, either triggering an event on their next page load, or popping up something like one of those embedded chats.