Nothing special would happen, as per the famous no-hair theorem, because the material that formed a black hole is irrelevant to an outside observer.
The no-hair theorem says that the three key properties of a black hole are its mass, charge, and angular momentum. When studying black holes, we don't care about the specific type of matter that formed them - protons, neutrons, neutrinos, photons, etc., whether they be matter or antimatter. The only important thing is that there is a certain amount of mass-energy contained within a certain volume. Therefore, a black hole formed from matter is indistinguishable from a black hole of the same mass, charge and angular momentum but formed from antimatter.
This means that nothing unusual would happen if the two black holes interacted or even merged into a more massive black hole. We can't really speculate about what happens beyond the event horizon, but if you had, say, a proton and an antiproton annihilate within the Schwarzschild radius, the black hole would remain unchanged - while energy would be released in the form of photons, those photons are still bound by the speed of light and would be unable to escape the black hole. It's a consequence of special relativity that changing mass into energy doesn't do anything.
One way you could see some sort of matter-antimatter reaction would be if the black holes have accretion disks, one of matter and one of antimatter. As the two bodies came closer and closer together, the disks would interact, leading to matter-antimatter annihilation and the emission of high-energy photons; electron-positron annihilation, for example, usually leads to the emission of gamma rays. This would lead to some interesting effects; the disks might disperse from the resulting radiation pressure.
This setup is admittedly a bit unrealistic. We don't expect matter and antimatter to exist in the same place in significant quantities, as they'd almost immediately annihilate. It really depends on whether you're willing to handwave anything.