  We analyze magnetic flux tubes at zero temperature in a superconductor that
is coupled to a superfluid via both density and gradient (``entrainment'')
interactions. The example we have in mind is high-density nuclear matter, which
is a proton superconductor and a neutron superfluid, but our treatment is
general and simple, modeling the interactions as a Ginzburg-Landau effective
theory with four-fermion couplings, including only s-wave pairing. We
numerically solve the field equations for flux tubes with an arbitrary number
of flux quanta, and compare their energies. This allows us to map the
type-I/type-II transition in the superconductor, which occurs at the
conventional kappa = 1/sqrt(2) if the condensates are uncoupled.
  We find that a density coupling between the condensates raises the critical
kappa and, for a sufficiently high neutron density, resolves the type-I/type-II
transition line into an infinite number of bands corresponding to
``type-II(n)'' phases, in which n, the number of quanta in the favored flux
tube, steps from 1 to infinity. For lower neutron density, the coupling creates
spinodal regions around the type-I/type-II boundary, in which metastable flux
configurations are possible. We find that a gradient coupling between the
condensates lowers the critical kappa and creates spinodal regions. These
exotic phenomena may not occur in nuclear matter, which is thought to be deep
in the type-II region, but might be observed in condensed matter systems.
