certain particles (i.e: certain bosons like the photon) do not have an anti-particle, or rather, they are they own anti-particles.
lets assume that such symmetry is only approximate and these particles are actually different than their antiparticles, just that the broken symmetry is very small and hard to detect
Question what experimental facts would we expect to change if such symmetries were only approximate? what physical consequences would they show?
EDIT The answers have focused on the argument that any symmetry breaking of this kind would be an all-or-nothing proposition. I would like to explore that argument a bit more with a more precise notion of a soft-broken symmetry, which would hopefully, elicit answers that will expose better the reason why symmetry breaking has to be an all-or-nothing:
what about photons hypothetically having a slightly higher coupling constant to matter than to antimatter, and a symmetric situation with their dagger counterparts, which would couple slightly stronger to anti-matter? what argument exists to discard that possibility theoretically? maybe we have bounds on the delta of such couplings? to what fraction of $\alpha$ are such bounds known experimentally?