The point of departure of the present paper is the observation that the strong fluctuation theory for statistically layered media needs not only the coordinate-acting effective permittivity operator (EPO) but also its spectral representation in the specific basis set of waves associated with this class of random media. Extending the author's previous work for isotropic and uniaxial media, the paper treats the EPO and its proper spectral counterpart for a statistically layered medium with arbitrary permeability and random permittivity dyads. These results have been obtained by exploiting a new recipe for regularizing the Green's dyad for the background medium. As compared with other approaches, the proposed one has the advantage of removing secular terms simultaneously in the expressions for the EPO and its spectral representation in the aforementioned basis set. In addition to the general case, the examples of gyrotropic random media and statistically rough interfaces separating deterministic layered media are considered.