Even though it’s been quite a few years since I attended any quantum mechanics courses, I did do a talk as an undergraduate on this very experiment, so I’m hoping that what I write below will not be complete rubbish. I’ll quickly go through the double slit experiment, and then try to explain what’s happening in the delayed choice quantum eraser and why it happens. Disclaimer: I know (or knew) the maths, but our professors did not go to great lengths explaining what ‘really’ happens, let alone what happens according to the MWI, so my explanation comes from my understanding of the maths and my admittedly more shoddy understanding of the MWI. So take the following with a grain of salt, and I would welcome comments and corrections from better informed people! (Also, the names for the different detectors in the delayed choice explanation are taken from the wikipedia article)
In the normal double slit experiment, letting through one photon at a time, the slit through which the photon went cannot be determined, as the world-state when the photon has landed could have come from either trajectory (so it’s still within the same Everett branch), and so both paths of the photon were able to interfere, affecting where it landed. As more photons are sent through, we see evidence of this through the interference pattern created. However, if we measure which slit the photon goes through, the world states when the photon lands are different for each slit the photon went through (in one branch, a measurement exists which says it went through slit A, and in the other, through slit B). Because the end world states are different, the two branch-versions of the photon did not interfere with each other. I think of it like this: starting at a world state at point A, and ending at a world state at point B, if multiple paths of a photon could have led from A to B, then the different paths could interfere with each other. In the case where the slit the photon went through is known, the different paths could not both lead to the same world state (B), and so existed in separate Everett branches, unable to interfere with each other.
Now, with the delayed choice: the key is to resist the temptation to take the state “signal photon has landed, but idler photon has yet to land” as point B in my above analogy. If you did, you’d see that the world state can be reached by the photon going through either slit, and so interference inside this single branch must have occurred. But time doesn’t work that way, it turns out: the true final world states are those that take into account where the idler photon went. And so we see that in the world state where the idler photon landed in D1 or D2, this could have occurred whether the photon went through either slit, and so both on D0 (for those photons) and D1/D2, we end up seeing interference patterns, as we’re still within a single branch, so to speak (when it comes to this limited interaction, that is). Whereas in the case where the idler photon reaches D3, that world state could not have been reached by the photon going through either slit, and so the trajectory of the photon did not interfere with any other trajectory (since the other trajectory led to a world state where the idler photon was detected at D4, so a separate branch).
So going back to my point A/B analogy, imagine three world states A, B and C as points on a page, and STRAIGHT lines represent different hypothetical paths a photon could take, you can see that if two paths lead from point A to point B, the lines would be on top of each other, meaning a single branch, and the paths would interfere. But if one of the paths led to point A and the other to point B, they would not be on top of each other, they go into different branches, and so the paths would not interfere.
Interesting questions to think about. Seeing if everyone independently describes the clothes the same way (as suggested by others) might work, unless the information is leaked. Personally, my mind went straight to the physics of the thing, ‘going all science on it’ as you say—as emperor, I’d claim that the clothes should have some minimum strength, lest I rip them the moment I put them on. If a piece of the fabric, stretched by the two tailors, can at least support the weight of my hand (or some other light object if you’re not too paranoid about the tailor’s abilities as illusionists), then it should be suitable.
Then, when your hand (or whatever) goes straight through, either they’ll admit that the clothes aren’t real, or they’ll come up with some excuse about the cloth being so fine that it ripped or things go straight through, at which point you can say that these clothes are useless to you if they’ll rip at the slightest movement or somehow phase through flesh, etc.
Incidentally, that’s one of my approaches to other things invisible to me that others believe in. Does it have practical uses or create a physical effect in the world? If not, then even if it’s really there, there’s not much point in acknowledging it...