It'd be useful to be able to define a task that doesn't start immediately. The task could start on some action / event, or when some other async process is done.

I could imagine two non-exclusive ways to do this:

1. Have a method to start a task, along with an option to not immediately run a task. That option could be a particular set of deps, as in [@lit-labs/task] Task function never call if there are no deps #2317. Maybe () => [] always runs the task, and undefined creates a manually run task.
2. Support Promises as deps and only run a task when the promises resolve. This could be tricky when dealing with Promise identity (new Promise instances wrapping the same underlying value) and needing to always wait a microtask to see the Promise value. Some of the same issues that Task solves vs until().