  We identify the possible ground states for a mixture of two superfluid
condensates (one neutral, the other electrically charged) using a
phenomenological Ginzburg-Landau model. While this framework is applicable to
any interacting condensed-matter mixture of a charged and a neutral component,
we focus on nuclear matter in neutron star cores, where proton and neutron
condensates are coupled via non-dissipative entrainment. We employ the Skyrme
interaction to determine the neutron star's equilibrium composition, and hence
obtain realistic coefficients for our Ginzburg-Landau model at each depth
within the star's core. We then use the Ginzburg-Landau model to determine the
ground state in the presence of a magnetic field. In this way, we obtain
superconducting phase diagrams for six representative Skyrme models, revealing
the microphysical magnetic flux distribution throughout the neutron star core.
The phase diagrams are rather complex and the locations of most of the phase
transitions can only be determined through numerical calculations. Nonetheless,
we find that for all equations of state considered in this work, much of the
outer core exhibits type-1.5 superconductivity, rather than type-II
superconductivity as is generally assumed. For local magnetic field strengths
$\lesssim 10^{14} \, {\rm G}$, the magnetic flux is distributed
inhomogeneously, with bundles of magnetic fluxtubes separated by flux-free
Meissner regions. We provide an approximate criterion to determine the
transition between this type-1.5 phase and the type-I region in the inner core.
