We propose an ideal experiment enabling the simultaneous assignment of the objective values, 0 or 1, of two incompatible properties of a system made up of two separated, non-interacting spin particles when a strict interpretation of the criterion of reality of Einstein, Podolsky and Rosen is adopted. We compare this experiment with the physical situation involving two-value observables of a system of two correlated spin-1/2 particles envisaged by Bohm; in particular, we show its inadequacy in the dual assignment at issue. (...) Finally, we discuss a number of questions of interpretation of the consistent quantum theory highlighted by our experiment. (shrink)