This paper is the second in a series that models photospheric magnetic structures in terms of atmospheres having optically thin fluctuations of magnetic field and thermodynamic state (the MISMA hypothesis). We apply an inversion procedure to the polarization of Fe I λ6301.5 and Fe I λ6302.5 observed in network and internetwork regions with the Advanced Stokes Polarimeter. Some 5200 independent spectra, comprising mildly asymmetric to very abnormal Stokes profiles, were reproduced by a single type of model atmosphere. It has three components, two that are magnetized and one that is not. A large fraction among the field strengths we measure are in the kG regime, but simulations suggest that the polarization of the observed Fe I lines weakens below detectable levels for fields substantially smaller than the observed ones. Synthesis of Stokes profiles of the IR Fe I λ15648.5 line in MISMAs reveals the opposite behavior, i.e., an increase of polarization for sub-kG fields. The highly transparent MISMAs inferred from observations are significantly brighter in the continuum than an unmagnetized atmosphere. The mass of the magnetic structures tends to be at rest, although a minor fraction undergoes strong downflows. Downflows are also present in the nonmagnetic environment. A significant number of fitted Stokes profiles require opposite magnetic polarities within the same resolution element. The occurrence of mixed polarities increases with weakening degree of polarization, such that 25% of the weakest signals require mixed polarity. The weak polarization signals account for most of the total (unsigned) magnetic flux of the observed region. By extrapolation, this indicates that a significant fraction of photospheric magnetic flux remains undetected. The MISMA framework provides a unified and physically consistent scenario for interpretation of quiet Sun magnetism. Moreover, it is the only one available at present that is able to fit the abnormal Stokes profiles as revealed by the new generation of sensitive Stokes polarimeters.