Well, first, that is not something that actually happens in the real world but is a misunderstanding. Particles diffract like a wave from a slit due to the uncertainty principle, because their position is confined to the narrow slit so their momentum must probabilistically spread out. If you have two slits where they have a probability of entering one slit or the other, then you will have two probabilistic diffraction trajectories propagating from each slit which will overlap with each other.

Measuring the slit the photon passes through does not make it behave like a particle. Its probabilistic trajectory still diffracts out of both slits, and you will still get a smeared out diffraction pattern like a wave. The diagrams that show two neat clean separated blobs has never been observed in real life and is just a myth. The only difference that occurs between whether or not you’re making a measurement is whether or not the two diffraction trajectories interfere with one another or not, and that interference gives you the black bands.

This is an interference-based experiment. Interference-based phenomena can all be given entirely classical explanations without even resorting to anything nonclassical. The paper “Why interference phenomena do not capture the essence of quantum theory” is a good discussion on this. There is also a presentation on it here.

Basically, you (1) treat particles as values that propagate in a field. Not waves that propagate through a field, just values in a field like any classical field theory. Classical fields are indeed something that can take multiple paths simultaneously. (2) We assume that the particles really do have well-defined values for all of their observables at once, even if the uncertainty principle disallows us from knowing them all simultaneously. We can mathematically prove from that assumption that it would impossible to construct a measuring device that simply passively measures a system, it will always perturb the values it is not measuring in an unpredictable way.

A classical field has values everywhere. That’s basically what a field is, you assign a value, in this case a vector, to every point in space and time. The vector holds the properties of the particles. For example, the X, Y, and Z observable would be stored in a vector [X, Y, Z] with a vector value at any point. What the measuring device measures is |0> or |1>, where we interpret the former to meaning no photon is there and we interpret the latter to mean a photon is there. But if you know anything about quantum information science, you know that |0> just means Z=+1 and |1> just means Z=-1. Hence, if you measure |0>, it doesn’t tell you anything about the X and Y values, which we would assume are also there if particles are excitations in a field as given by assumption #1 because the field exists everywhere, and in fact, from our other assumption #2, your measurement of its Z value to be |0> must perturb those X and Y values.

It would be the field that propagates information through both slits and the presence of the measurement device perturbs the observables you do not measure, causing them to become out of phase with one another so they that they do not interfere when the field values overlap.

Interestingly, this requires no modification to quantum mechanics. If a system is physically redundant, we can often ignore parts of it in the mathematics to simplify our calculations, but if we do so, then the mathematics don’t directly reflect the physical character of the system because parts of it are ignored. All we have to do is assume that for these kinds of photon-based and interference-based experiments that we are making a mathematical simplification due to redundancies and then can mathematically expand the description where it is more clearly obvious what is going on, and doing so is mathematically equivalent as it leads to the same predictions and, if you simplify it, it would lead to the same traditional way of describing the experiment.

It’s sort of like if you have 4, you can expand it into 2+2. It means the same thing, but 4 and 2+2 have physically different meaning, because 2+2 suggests two separate things coming together, whereas 4 suggests only 1 thing. Expanding the double-slit experiment is a bit complicated because position is continuous, but it’s trivial to demonstrate it for something like the Mach-Zehnder interferometer. You just map |0> to |01> and |1> to |10>, and then all the paradoxes with that, including the “bomb tester” paradox, disappear.