Antimatter is composed of antiparticles which possess the same mass as normal particles, but opposite electrical charges. Antimatter behaves almost exactly as normal matter does.
We would look at this binary star system with our detectors (telescopes in general). If we could look at the orbits of this binary system, I expect we won't find any evidence, since it is quite expected (but not confirmed experimentally) that antimatter interacts gravitationally the same way as normal matter does because they have the same mass.
Since antimatter acts almost exactly as normal matter does, observing the light emitted by the stars would result in no conclusions. So our best hope would be to wait for this antimatter to interact with normal matter. We know that space is filled with normal matter, so it is likely that some normal matter would get in contact with these stars and particle-antiparticle annihilation would occur, releasing the maximum energy possible for a given amount of mass involved in the process. By looking at the energy released by this reaction we could deduce that the stars are made up of antimatter.
Other than that, it would be almost impossible to test his theory. Such binary systems are very unlikely to exist, due to the amount of normal matter that we see in the universe. If such amount of antimatter were to exist, it would rapidly annihilate itself with normal matter, creating fireworks of energy in the sky that we would be able to detect.