In an effort to work out a believable form of faster-than-light communications for my setting, I've ruled out a few things: wormholes (which are in use, but are too random in where they lead, to be used for this) and quantum entanglement (which flat out doesn't work for information transfer).

In my research, the only natural phenomenon that even hints at being capable of this is the hypothetical tachyon particle. The problem is, they're believed to not be capable of interacting with anything, if they do exist.

Does anyone possibly have a suggestion for a minimal-handwaving way to acceptably explain away how they might detect these, in light of this issue?

Minimal handwaving is done with a single big lie rather than little ones.

If you try to tell the little lie, there's constant follow up, but what about this, but what about that, but what about the other.

The big lie tells not about the technology but about the timeline, not about how it was done but about who did it. Make it a story about the person who did it, where they were in their career, what country they were in. The war they'd just survived, when they emigrated from small war-torn country to large technological nation with nothing to their name. Acceptance to legendary institution, the great breakthrough, the implications and celebrations, but never how the technology actually works.

That way you have to wave your hand precisely once, rather than over and over again in lots of little ways.

Tachyons are detectable.

Fortunately, I believe your question is based on a mistaken premise. Tachyons, if they exist, would likely indeed be detectable. In fact, since they were initially theorized, there have been several experimental searches for tachyons, though very few in recent years. I'll talk about a couple experiments noted in Status of experimental searches for tachyons? They're important because they represent a couple main avenues of detection, and you should be able to base your communication system on them.

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  Clay 1988: Cosmic ray showers. As high-energy particles plow through the upper atmosphere, they decay into a number of lighter particles which are detectable by humans on Earth. The products of the decay of the initial particle all travel extremely close to the speed of light. Clay notes that the first scientist to exploit this phenomenon was Ramana Murthy, who proposed looking for particles that arrive even sooner, before the first photons from the event.


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  Alvager & Kreisler 1968: Cherenkov radiation. The effective speed of light is different in different mediums; photons interact with atoms in particles in a substance, effectively slowing them down. If a charged massive particle travels faster than this effective speed of light, it should emit photons called Cherenkov radiation, which is a well-studied phenomenon for massive particles. As tachyons travel faster than the speed of light in a vacuum (and thus faster than the speed of light in any medium), they should produce Cherenkov radiation, and would in fact be the only particles to produce such a signal in a vacuum.


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  Alvager & Erman 1965: Mass-energy and momentum. We can use special relativity to calculate the magnitude of a particle's energy $E$ and momentum $p$. For normal massive particles, we expect $|E|>|p|$; for tachyons, we should see $|p|>|E|$. I am still trying to find out more details of their experiment; the pair monitored an isotope of thulium, $^170textTm$. Thulium-170 usually transitions to Ytterbium-170 via $beta^-$ decay, but it appears that tachyons could play a role in more complicated processes.


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  Baltay et al. 1970: Missing energy. Even in the case of tachyons that don't interact (or don't interact strongly) with detectors, we should still be able to see them indirectly. In particular, some unstable particles might have decay chains involving tachyons, and if these chains are observed and some energy remains unaccounted for, it could be a sign of tachyons. Neutrinos, incidentally, were originally detected basically the same way.

The basic point is, tachyons can be detected directly (e.g. as products of cosmic rays or atomic decays) and indirectly (e.g. through Cherenkov radiation and missing energy from meson decay).

Applying this to communication

These experiments are, to be frank, not very useful for communication. Most involve observing tachyons produced naturally, instead of by humans, and at effectively uncontrollable rates. We can rule out most of them for your use, but I think the most promising is Alvager & Kreisler's method of Cherenkov radiation. Let me talk about their idea in slightly more detail.

The pair's setup involved two parallel plates with a static electric field between them. Tachyons should gain energy traveling through the field while losing energy via radiation, and it should be possible to tweak the field's parameters such that this total energy changer is zero - which they did. The tachyon should, over the course of traveling through the detector, travel through a potential difference of $sim9text kV$ and gain corresponding energy based on its charge; the Cherenkov photons would have energies in the range $0text-3.8text eV$. It was expected that 12% of the produced photons would be detected (although no doubt we could, today, increase that percentage). Tachyons with charges from $0.1text-2e$ could be observed.

^{Figure 1, Alvager & Kreisler 1968. A diagram of the duo's detector.}

I would guess that this setup could be scaled up such that detecting tachyons traveling over interstellar distances would be feasible. Presumably, information would be coded in the number of tachyons detected, and therefore the amount of energy produced in the form of photons. Furthermore, of course, you are perfectly able to change the parameters of the device and the properties (e.g. charge) or your tachyons, so you can optimize the process as you wish.

Suspension of disbelief, handwaving, and all that

Separatrix's answer, which I think also makes your assumption of undetectability, argues that you should do as little handwaving as possible - quite true - by avoiding discussing the details of the technology. This can be quite effective, and it definitely should not be ignored. I could stand to make use of it more.

That said, the basic idea behind the Alvager & Kreisler detector is simple enough that I believe this issue is not very important. If you wish to go slightly in depth when describing the device - or if you want one character to use something akin to jargon while talking to another - simply mention the electric fields used, or the potentials, or the energy range. I'm not a fan of using random (and irrelevant) jargon in writing, but in this case, it's quite relevant indeed, and the detector is simple enough that you're not as likely to alienate readers as you might think.

Mary particle.

Tachyons always move faster than light, and so also backwards in time. It is hard to imagine how to interact with something like that from our standpoint in the kitchen with a coffee. Sort of like interacting with God. God is so Godly.

But what about things like massless particles - our familiar friend the photon, and his weird cousins gluon and graviton? Those things always move at the speed of light. What does a tachyon look like from the perspective of a massless particle? From that perspective what is the tachyon up to? Mary is familiar and motherly, and less intimidating than God - the glorious intermediary. So too your particle.

In your fiction, you can discover that tachyons can be detected through their (time-backwards) interactions with massless particles. To keep it squarely in fiction you can invoke the little known graviton since it is so mysterious you can assert what you like. People might call you out if you ascribe new properties to the photon. But you could.

It hurts my head some to think about what interactions between a light speed (?timeless) particle and a superluminal time backwards particle would look like. Good luck!

Tachyons aren't known to exist. But if they did, they could absolutely interact with conventional matter. That being said, you'd probably have trouble sending tachyon pulses around the galaxy. Not because of any tachyon-specific issues, but just because galaxies are huge. Right now there are probes about 20 million miles from Earth, and picking up their radio signals is a huge to-do. A galaxy is a billion times bigger than that distance, which means that the energy is diffused a billion billion (a quintillion) times more.

Depending on your setting, it might be reasonable to send the message to the nearest outpost, and wormhole it to some other outpost, and then tachyon it to your friend.