And, because this is the sort of person that I am, I thought: "what if you set out to do the exact opposite? A film that was normal in most respects, but in which every point of concern that a continuity expert might attend to were perverted and made to be 'exactly wrong'?" I'm going for the "uncanny valley" effect where things are just normal enough to create the expectation of normalcy but pull out the rug from under it *continuously*.
I think it would be important for the characters to maintain their full narrative arc, and pay no mind to any of the discontinuities, yet be sources of it at the same time. Not so crazy as to become unparseable, but enough to keep disbelief suspended on the edge of a chair tipped back too far. (Negativland comes to mind, but I haven't seen them going for this effect exactly; what I've seen their stuff has been either overlapping video loops as a sort of visual percussion section, or a series of still images.)