That's a head-scratcher, Dr. H.
Try some different combinations to see if you can narrow down the circuit that's the culprit. Does it do it if your engine switches are all "on" but you haven't started the engine yet? What about isolating one ignition and then the other and seeing if the strobes work? Can you get a helper to wiggle the wires with the engine running and then not running to see what happens? Not sure if those make sense, but maybe it will help to zero in on the issue.
I'm kind of leaning toward what the others are saying about grounds. I've seen them cause all kinds of strange problems that don't make apparent sense--both at work and on planes & cars.
Let us know what you find.