Properly plumbed, there should not be any connection between the fresh water plumbing and the black tank flush or tank itself. The black tank flush is a dedicated connection and line to the flusher itself. And gray water should not enter the black tank. Likewise there should not be any connection between the black and gray tanks, except at the sewer drain connection. There was a time when some manufacturers would connect a gray water drain to the black tank, but today that violates the RVIA standards.
Could someone have modified the plumbing to incorporate a direct connection between the two; it is possible but certainly not advisable. I think you mentioned you bought the unit new. Could it have been mistakenly connected somehow when built, also possible, but then the dedicated flushing connection might not work.
A very interesting conundrum, I will be watching to see if you get any answers.
2014 Seneca 37TS
Toads: 2014 Ford Explorer Sport or 2013 Ford Focus ST