Additional variables (also often called “hidden variables”) are sometimes added to standard quantum mechanics in order to remove its indeterminism or “incompleteness” and to make the measurement process look more classical. Here we discuss a case in which an additional variable arises almost spontaneously from the quantum formalism: the emergence of a relative phase between two highly populated Fock-state Bose-Einstein condensates. The model simulated here involves the interference of two Bose condensates, one with all up spins and the other with down spins, along a z axis. With the clouds overlapping, we consider the results of measuring spins in a transverse plane (the general direction is studied in an appendix). The determination of the previously “hidden” phase becomes progressively more definite as additional measurements are made. We also provide an analysis of a recent and closely related experiment.