  We study theoretically the properties of two Bose-Einstein condensates in
different spin states, represented by a double Fock state. Individual
measurements of the spins of the particles are performed in transverse
directions, giving access to the relative phase of the condensates. Initially,
this phase is completely undefined, and the first measurements provide random
results. But a fixed value of this phase rapidly emerges under the effect of
the successive quantum measurements, giving rise to a quasi-classical situation
where all spins have parallel transverse orientations. If the number of
measurements reaches its maximum (the number of particles), quantum effects
show up again, giving rise to violations of Bell type inequalities. The
violation of BCHSH inequalities with an arbitrarily large number of spins may
be comparable (or even equal) to that obtained with two spins.
