One more thing...
I also tried resetting the the USB device, both under software control and manually asserting the FTDI reset line when the problem occurred.
This was done in the hope that I could achieve similar results (i.e. no issue afterwards) to the shorting of DP/DM.
In both cases the FTDI device disconnected and re-enumerated but the problem still persists afterwards.
So there seems to be something good happening specifically, and only with shorting DP/DM.
What could that be?