Gravitational waves from astrophysical sources can interact with background electromagnetic fields, giving rise to distinctive and potentially detectable electromagnetic signatures. In this paper, we study such interactions for far-field gravitational radiation using the 1+3 approach to relativity. Linearised equations for the electromagnetic field on perturbed Minkowski space are derived and solved analytically. The inverse Gertsenshtein conversion of gravitational waves in a static electromagnetic field is rederived, and the resultant electromagnetic radiation is shown to be significant for highly magnetised pulsars in compact binary systems. We also obtain a variety of nonlinear interference effects for interacting gravitational and electromagnetic waves, although wave-wave resonances previously described in the literature are absent when the electric-magnetic self-interaction is taken into account. The fluctuation and amplification of electromagnetic energy flux as the gravitational wave strength increases towards the gravitational-electromagnetic frequency ratio is a possible signature of gravitational radiation from extended astrophysical sources.