The interference of two Bose-Einstein condensates, initially in Fock states, can be described in terms of their relative phase, treated as a random unknown variable. This phase can be understood, either as emerging from the measurements, or preexisting to them; in the latter case, the originating states could be phase states with unknown phases, so that an average over all their possible values is taken. Both points of view lead to a description of probabilities of results of experiments in terms of a phase angle, which plays the role of a classical variable. Nevertheless, in some situations, this description is not sufficient: another variable, which we call the "quantum angle", emerges from the theory. This article studies various manifestations of the quantum angle. We first introduce the quantum angle by expressing two Fock states crossing a beam splitter in terms of phase states, and relate the quantum angle to off-diagonal matrix elements in the phase representation. Then we consider an experiment with two beam splitters, where two experimenters make dichotomic measurements with two interferometers and detectors that are far apart; the results lead to violations of the Bell-Clauser-Horne-Shimony-Holt inequality (valid for local-realistic theories, including classical descriptions of the phase). Finally, we discuss an experiment where particles from each of two sources are either deviated via a beam splitter to a side collector or proceed to the point of interference. For a given interference result, we find "population oscillations" in the distributions of the deviated particles, which are entirely controlled by the quantum angle. Various versions of population oscillation experiments are discussed, with two or three independent condensates.