I've come up with a new solution, which has a tiny bit of overhead, but seems to be working so far as a prototype. One assumption is that you're in an honour system environment for logging in, although this could be adapted by rerequesting a password whenever you switch tabs.
Use localStorage (or equivalent) and the HTML5 storage event to detect when a new browser tab has switched which user is active. When that happens, create a ghost overlay with a message saying you can't use the current window (or otherwise disable the window temporarily, you might not want it to be this conspicuous.) When the window regains focus, send an AJAX request logging the user back in.
One caveat to this approach: you can't have any normal AJAX calls (i.e., ones that depend on your session) happen in a window that doesn't have the focus (e.g. if you had a call happening after a delay), unless you manually make an AJAX re-login call before that. So really all you need do is have your AJAX function check first to make sure localStorage.currently_logged_in_user_id === window.yourAppNameSpace.user_id, and if not, log in first via AJAX.
Another is race conditions: if you can switch windows fast enough to confuse it, you may end up with a relogin1->relogin2->ajax1->ajax2 sequence, with ajax1 being made under the wrong session. Work around this by pushing login AJAX requests onto an array, and then onstorage and before issuing a new login request, abort all current requests.
The last gotcha to look out for is window refreshes. If someone refreshes the window while you've got an AJAX login request active but not completed, it'll be refreshed in the name of the wrong person. In this case you can use the nonstandard beforeunload event to warn the user about the potential mixup and ask them to click Cancel, meanwhile reissuing an AJAX login request. Then the only way they can botch it is by clicking OK before the request completes (or by accidentally hitting enter/spacebar, because OK is--unfortunately for this case--the default.) There are other ways to handle this case, like detecting F5 and Ctrl+R/Alt+R presses, which will work in most cases but could be thwarted by user keyboard shortcut reconfiguration or alternative OS use. However, this is a bit of an edge case in reality, and the worst case scenarios are never that bad: in an honour system configuration, you'd be logged in as the wrong person (but you can make it obvious that this is the case by personalizing pages with colours, styles, prominently displayed names, etc.); in a password configuration, the onus is on the last person who entered their password to have logged out or shared their session, or if this person is actually the current user, then there's no breach.
But in the end you have a one-user-per-tab application that (hopefully) just acts as it should, without having to necessarily set up profiles, use IE, or rewrite URLs. Make sure you make it obvious in each tab who is logged into that particular tab, though...