  We discuss the properties of two Bose-Einstein condensates in different spin
states, represented quantum mechanically by a double Fock state. Individual
measurements of the spins of the particles are performed in transverse
directions (perpendicular to the spin quantization axis), giving access to the
relative phase of the two macroscopically occupied states. Before the first
spin measurement, the phase is completely undetermined; after a few
measurements, a more and more precise knowledge of its value emerges under the
effect of the quantum measurement process. This naturally leads to the usual
notion of a quasi-classical phase (Anderson phase) and to an interesting
transposition of the EPR (Einstein-Podolsky-Rosen) argument to macroscopic
physical quantities. The purpose of this article is to discuss this
transposition, as well as situations where the notion of a quasi-classical
phase is no longer sufficient to account for the quantum results, and where
significant violations of Bell type inequalities are predicted.
