We investigate the invisibility via anomalous localized resonance of a general source in anisotropic media for electromagnetic waves. To this end, we first introduce the concept of doubly complementary media in the electromagnetic setting. These are media consisting of negative-index metamaterials in a shell and positive-index materials in its complement for which the shell is complementary to a part of the core and a part of the exterior of the core-shell structure. We then provide criteria for establishing the invisibility of a source in these media. We show that (i) a source is invisible if the power is blown up; (ii) a source is invisible if it is sufficiently close to the plasmonic shell structure, and it is visible if it is far from this plasmonic structure; (iii) if the plasmonic structure is complementary to an annulus of constant isotropic medium, there is a critical length that characterizes the cloaking phenomena, as first observed in the two-dimensional acoustic quasistatic setting by Milton and Nicorovici.