In composite materials, with field-dependent restructuring of the filler material (changes in the mutual arrangement of inclusions), the presence of an external magnetic field induces anisotropy of the dielectric properties, even if the composite is isotropic in the absence of an external field. A modified effective medium approximation is proposed for the calculation of the components of effective permittivity within a class of composites with reconfigurable microstructure, where both phases (the filler and the matrix) are isotropic and the inclusions have spherical shape. The effective physical properties are calculated in the parallel and perpendicular directions to an applied field. The appearance of the anisotropy of the permittivity is simulated by the introduction of two not-equal, possibly variable (field-dependent) percolation thresholds. The implications, of the proposed theoretical approach, are demonstrated for the case of the dielectric properties of magnetoactive elastomers (MAEs). In MAEs with soft polymer matrices, the mutual arrangement of micrometer-sized magnetic inclusions can significantly change in an applied magnetic field. A reasonable agreement between theory and experiment at a measurement frequency of 1 kHz is found, and is improved in comparison to the previous models. The components of the effective permittivity tensor, characterizing the dielectric properties along the direction of the applied magnetic field and in the orthogonal direction, grow with an increasing field. This growth is more pronounced for the permittivity component in the field direction. The possible extensions of the theoretical model and future directions of research are discussed. The presented theoretical approach can be useful for the application-driven development of a number of smart materials, in particular electro- and magnetorheological gels, elastomers and fluids. •The mechanism of the field-induced anisotropy of the dielectric properties of magnetoactive elastomers is proposed.•In order to describe the anisotropy, the effective medium approximation is modified.•The calculated dependence of the permittivity tensor on a magnetic field is compared with the experiment.