Depends on D20654. Ref T4900. When a task is edited, emit a "workboards" event for all boards it appears on (in a future change, this should also include all boards it previously appeared on, and all parents of both sets of boards -- but I'm just getting things working for now).
When we receive a "workboards" event, check if the visible board should be updated.
Aphlict has a complicated intra-window leader/follower election system which could let us process this update event exactly once no matter how many windows a user has open with the same workboard. I'm not trying to do any of this since it seems fairly rare. It makes sense for events like "you have new notifications" where we don't want to generate 100 Ajax calls if the user has 100 windows open, but very few users seem likely to have 100 copies of the same workboard open.