Consider a riff on a Michelson-style interferometer that looks like this:
A source of laser light shines on a half-silvered mirror angled at 45 degrees (the grey rectangle). This splits the beam in two. The two beams are in actuality the same color as the original, but I've drawn them in two different colors to make the two paths easier to follow. The red beam is reflected up, the green beam passes through the mirror and continues to the right. Both beams are reflected back from whence they came by a pair of regular mirrors (the white rectangles) also mounted at 45 degree angles relative to the beams. They return to the half-silvered mirror where they are recombined and sent to a detector, which registers the presence of absence of interference. (In reality it's a tad more complicated than that, but that's a sufficiently accurate description for this thought experiment.)
What distinguishes this "riff" on a traditional Michelson interferometer is that the mirrors that reflect one of the beams (the one drawn in green) are mounted on a trolley that allows them to be moved as a unit to an arbitrary distance. We use this capability to make the distance traversed by the photons on the green path be much larger than those on the red path, large enough that there is an easily measured difference in the travel time between the two paths.
Let's call the time it takes to traverse the red path T1 and the time it takes to traverse the green path T2, with T1 much smaller than T2.
So we turn on the laser. What can we expect to see? Well, the laser beam is emitted at the speed of light (obviously). After time T1 the red-path photons arrive, but the green-path photons are still en-route, so we should see no interference. Then at time T2 the green-path photons arrive. What happens then?
From a purely electromagnetic point of view we would now expect to see interference. But do we? Here is an argument that this cannot be the case: whether we see interference or not should be independent of the brightness of the beam. So turn the brightness down to the point where the time between the emission of individual photons from the laser (and hence the arrival of photons at the detector) is much greater than T2. (Of course, the actual arrival times will be random, but we can make the average time between photons be large enough that the probability of having two photons in the interferometer at the same time is indistinguishable from zero.)
If we do this with a standard interferometer (or two-slit experiment) where the path-lengths are very nearly the same, we still see interference even when the photons go through one at a time. This is the famous and mysterious phenomenon of quantum superposition, where each individual photon "goes both ways" and interferes with itself. But with this setup, "interfering with itself" would seem to be impossible. Yes, the photon goes both ways, but how can it possibly interfere with itself when the differences in travel times between the two paths are so large? By the time the red-path-part of the photon arrives at the detector, the green-path part is still en-route to the distant mirror. Likewise, by the time the green-path-part of the photon arrives at the detector, the red-path part is long gone. So individual photons can't possibly interfere with themselves in this setup, and so large numbers of photons should not be able to produce interference either.
So there are three possibilities:
1. There is no interference after T2 (in violation of standard electromagnetic theory)
2. There is interference after T2, but it goes away if the laser is dim enough (in violation of standard QM theory), or
3. There is interference after T2 even when the laser is dim. In which case the question becomes: how?
The answer next time. It turns out that this thought experiment has some pretty profound implications with respect to the interpretation of quantum mechanics. As far as I can tell from a cursory search, I'm the first to propose it, though I would be surprised if that actually holds up to scrutiny. If anyone knows where this has been analyzed in the literature I'd appreciate a pointer.