How to keep asynchronous parallel program code manageable (for example in C++)?

You could consider an explicit "event loop" or "message loop", not too different from classic approaches such as a select loop for asynchronous network tasks or a message loop for a windowing system. Events that arrive may be dispatched to a callback when appropriate, such as in your example B, but they may also in some cases be tracked differently, for example to cause transactions in a finite state machine. A FSM is a fine way to manage the complexity of an interaction along a protocol that requires many steps, after all!

I tend to go with B, but instead of calling forth and back, I'd do the entire processing including follow-ups on a separate thread. The main thread can meanwhile update the GUI and either actively wait for the thread to complete (i.e. Show a dialog with a progress bar), or just let it do its thing in the background and pick up the notification when it's done.

No complexity problems so far, since the entire processing is actually synchronous from the processing thread's point of view. From the GUI's point of view, it's asynchronous.

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