Complementarity and the pathological statistics of the quantum impossible

Standard quantum physics prevents the existence of a joint statistics for complementary observables. Nevertheless, a joint distribution for complementary observables can be derived from their imperfect simultaneous measurement followed by a suitable data inversion. Quantumness is reflected then in the pathologies of the inferred distribution. We apply this program to the paradigmatic example of complementarity: the wave-particle duality in a Young interferometer.