Unfortuantely I cant' really say you support real-time updating on stock levels (or any other resources, actually). Even without going into the multiple tab issue, even with a single user, single tab, I end up refreshing the page all the time. Whenever something changes (a new order comes, a customer with a new address has placed an order...) navigating between the UI never update.
I can see that you guys are using EmberJS, which is awesome, but it seems you're being way too aggressive in the way you cache things. If you enter into the "Orders" page, it loads the orders once. Then, navigating into other parts of the app, and then going back to the Orders, you do not do any new API calls, and instead re-use the orders that you may have loaded minutes ago, so obviously this is going into out of sync.
I understand that it will likely increase the number of API calls you'll do, but please make sure that going into a page reload the data from the source of truth (your API). I know you are using both EmberJS and Ember-Data, I'm a big fan of those two libraries, and I know how easy it is to force refresh the models...