For quarks, leptons, and even composite particles, a real particle and corresponding antiparticle can annihilate each other. Is there a similar, or analogous interaction between bosons of the same type given that bosons are 'their own antiparticles'?
A Google search for "boson boson interaction" didn't include many results about bosons of the same type interacting. I did find this link but it hasn't loaded yet for me:
First, bosons aren't generally their own antiparticles. Hydrogen is a boson, and can annihilate with antihydrogen, but isn't identical with antihydrogen. It's also possible for fermions to be their own antiparticles, e.g. the right-handed neutrinos in many popular Standard Model extensions.
There aren't any special rules governing annihilations. The rule is that any interaction that doesn't violate any conservation laws will happen with some nonzero amplitude (the "totalitarian principle"). In relativistic quantum field theories, there's also a theorem that antiparticles exist with opposite values of all conserved quantities except energy-momentum. That means that if $a$ is some particle and $b$ is another particle with lower (perhaps zero) rest mass, then $a+\bar a \to b+\bar b$ (where the overbar denotes antiparticle) is always allowed. This includes, for instance, $2Z^0 \to 2γ$ or $2π^0 \to 2γ$ which could be called "self-annihilation" of identical bosons. (The $Z^0$ and $γ$ aren't directly coupled, but the totalitarian principle correctly says that the interaction can happen anyway – it's just not very likely.)
A collision of two bosons of the same type which produces something else entirely is certainly allowed. But that would usually not be called "annihilation" unless the final state particles are massless. The massless particles in the standard model all come from gauge fields and this is what tightly constrains the process you're imagining.
Can two bosons of the same type annihilate into gluons?
They could only do this if they coupled to the $SU(3)$ gauge field which would imply that they had color. A fairly large energy like $~500 \mathrm{MeV}$ is needed for deconfined quarks and the experiment would become even more difficult if we wanted to produce a composite particle that was bosonic and colored. However, it should be possible in principle. Note that gluon final states would quickly produce jets and look like something massive again.
Can two bosons of the same type annihilate into photons?
They could only do this if they coupled to the $U(1)$ gauge field which would imply that they had charge. But then they would of course need opposite charge and any experimentalist would be justified in calling the positively charged one the anti-particle of the other. So here we can say that bosons of the same type by definition cannot annihilate into photons if they are their own anti-particles.
The most instructive toy model for seeing bosons which behave analogously to electrons and positrons in this respect is scalar QED.
