How can we detect antihydrogen?

As you've already noted, we cannot detect cosmic antimatter from its spectrum.

As the answers at How would we tell antimatter galaxies apart? indicate, there are two ways we could detect cosmic antimatter.

Firstly, we would see the tell-tale 511 keV gamma ray signature of electron + positron annihilation reactions coming from the border of the antimatter region where it contacts normal matter. Antiprotons & antineutrons will (of course) also annihilate on contact with normal matter. Such annihilation reactions are rather messy, and can emit gamma rays of various energies. They also emit other particles, eg pions, but they decay quickly, and the long-distance signature of these reactions is fairly similar to that of the electron + positron annihilation. I give more details on annihilation reactions in my answer to What happens to the quantum information of a particle and an antiparticle when they annihilate?

Annihilation reactions are intense. After all, all the mass involved gets converted to photons, whereas even the most powerful nuclear fusion reactions only convert a small percentage of rest mass to photons & kinetic energy. But you can't compare annihilation reactions at the border of a space gas cloud with the power of a supernova.

Space gas tends to be extremely tenuous, with fewer particles per cubic metre than the hardest vacuums that we can produce on Earth. A cloud of antihydrogen reacting with the surrounding hydrogen would give off a lot of gamma, heating the gas, but it doesn't create any kind of chain reaction, and the cloud will take some time to completely annihilate.

According to Hydrogen-Antihydrogen Collisions [P. Froelich, S. Jonsell, A. Saenz, B. Zygelman, and A. Dalgarno Phys. Rev. Lett. 84, 4577 – Published 15 May 2000]

Surprisingly, starting with $n_H = n_{\bar H} = 10^7 \text{cm}^{-3}$ and energy <10 K, it takes a whole 17 minutes for the mixture of equal amounts of hydrogen and antihydrogen to lose half of all atoms.

$n_H$ and $n_{\bar H}$ are the particle number densities of hydrogen and antihydrogen, respectively, in a cold homogeneous mixture of hydrogen and antihydrogen. Note that particle densities in the interstellar medium are typically much smaller, ranging from $20 \text{cm}^{-3}$ to $10^{-4} \text{cm}^{-3}$. The mean density in molecular clouds, where stars are formed, can be much higher, eg $10^2 – 10^6 \text{cm}^{-3}$, but even the dense end of that range is 1/10 the density mentioned in that quote from Froelich, Jonsell et al.

OTOH, the collision of a normal star with an antimatter star would be extremely spectacular. ;)

The other option mentioned at the linked question is that if there are whole stars made of antimatter we might be lucky enough to detect the antineutrinos from an antimatter core collapse supernova. As explained on Wikipedia, core collapse also produces large amounts of thermal neutrinos & antineutrinos in equal quantities, and they outnumber the electron capture neutrinos by several times. So we'd need to detect both neutrinos and antineutrinos and measure their ratio carefully to distinguish between a regular supernova and an antimatter one.

Neutrino / antineutrino detection is hard. The best detectors using current technology can only detect neutrinos with kinetic energy around 300,000 times their rest mass. And even then, billions of neutrinos pass straight through the detector undetected for each neutrino that is detected. We did detect a handful of neutrinos from the supernova SN 1987A in the Large Magellanic Cloud). Hopefully, modern detectors could detect a few more, if the supernova were close enough. But I don't know if we could catch enough of them to make a sufficiently accurate determination of their ratio.

Antihydrogen atoms have been created in the lab and their basic spectral characteristics confirmed as identical to hydrogen. So we cannot tell by directly observing an object.

But we infer it from the fact that no gross interactions between matter and antimatter have been observed. A full answer is given at How would we tell antimatter galaxies apart?, but here is a short version.

Interstellar space, even intergalactic space, is not entirely empty. Matter may be vastly attenuated, down to a handful of atoms per cubic metre or whatever, but it is still there. If an object were made of antimatter, some of its atoms would dissipate into space and would eventually meet ordinary matter. Some matter in space is ionised. Electrons and anti-electrons (positrons) are oppositely charged. Lab experiments have shown that they will briefly bond to form a positronium atom before mutually annihilating themselves in a flash of radiation. Something similar must occur between a proton and an antiproton, though I do not know if this has been lab-tested. Other, charge-neutral annihilation interactions will also occur.

So if antimatter were out there, there would be a constant trickle of the characteristic radiation from the buffer zone, with occasional mega-bursts as material and anti-material objects collided. These radiation emissions would be readily detectable with modern instruments, but they are simply not there in the sky.

Quite why there is no antimatter out there is one of life's little mysteries; something somewhen broke a symmetry (conservation law), but we have no idea what, when or how.