I don't mind that the data would be stale, but it sounds like a significant architectural change, and I don't feel entitled to it!
I've actually spent a ridiculous amount of time tweaking the searches for my personal GTD system to get them to run as fast as possible, using many different approaches, and I'm confident I've hit the limits of that type of optimization. (I have a background as a SQL developer.)
So, it would still be very useful, despite the potential of showing stale data. Knowing that navigating away from a tab will result in having wait for search results to load again makes me generally hesitant to move around the interface as freely as I otherwise would, or build queries as sophisticated.
As always, many thanks for your response, and for the great software!
Cheers, Aaron