The interaction of streamers in nitrogen-oxygen mixtures such as air is studied. First, an efficient method for fully three-dimensional streamer simulations in multiprocessor machines is introduced. With its help, we find two competing mechanisms how two adjacent streamers can interact: through electrostatic repulsion and through attraction due to nonlocal photoionization. The nonintuitive effects of pressure and of the nitrogen-oxygen ratio are discussed. As photoionization is experimentally difficult to access, we finally suggest to measure it indirectly through streamer interactions.