  Additional variables (also often called ``hidden variables'') are sometimes
added to standard quantum mechanics in order to remove its indeterminism or
``incompletness,'' 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 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.
