  We consider the Berlin-Kac spherical model for supercritical densities under
a periodic lattice energy function which has finitely many non-degenerate
global minima. Energy functions arising from nearest neighbour interactions on
a rectangular lattice have a unique minimum, and in that case the supercritical
fraction of the total mass condenses to the ground state of the energy
function. We prove that for any sufficiently large lattice size this also
happens in the case of multiple global minima, although the precise
distribution of the supercritical mass and the structure of the condensate mass
fluctuations may depend on the lattice size. However, in all of these cases,
one can identify a bounded number of degrees of freedom forming the condensate
in such a way that their fluctuations are independent from the rest of the
fluid. More precisely, the original Berlin-Kac measure may be replaced by a
measure where the condensate and normal fluid degrees of freedom become
independent random variables, and the normal fluid part converges to the
critical Gaussian free field. The proof is based on a construction of a
suitable coupling between the two measures, proving that their Wasserstein
distance is small enough for the error in any finite moments of the field to
vanish as the lattice size is increased to infinity.
