In high-frequency magnetic structures (like thin-film magnetic heads) which have been laminated to avoid eddy current loss, it is standard to assume that electromagnetic coupling across the dielectric separating the magnetic layers is small. We show that for macroscopic films with insulating interlayers, the laminated structure is better modeled by an anisotropic effective medium with a magnetic permeability of around 3000 but an insulating dielectric function perpendicular to the lamination plane, while still metallic parallel to the lamination plane. Electromagnetic waves can propagate parallel to the lamination direction but with 1/100 of the free-space wavelength. The electromagnetic (capacitive) coupling between the layers can lead to radically altered propagation of magnetic flux and generate novel permeance resonances caused by interference effects across the width of the film. A Fourier series solution constructed with a mean-field theory for the wave equation for laminated slab geometries predicts the permeance resonances observed experimentally.