  The interplay between spontaneously broken gauge symmetries and Bose-Einstein
condensation has long been controversially discussed in science, since the
equation of motions are invariant under phase transformations. Within the
present model it is illustrated that spontaneous symmetry breaking appears as a
non-local process in position space, but within disjoint subspaces of the
underlying Hilbert space. Numerical simulations show that it is the symmetry of
the relative phase distribution between condensate and non-condensate quantum
fields which is spontaneously broken when passing the critical temperature for
Bose-Einstein condensation. Since the total number of gas particles remains
constant over time, the global U(1)-gauge symmetry of the system is preserved.
