  We describe and realize an experimental procedure for assessing the
incompatibility of two qubit measurements. The experiment consists in a state
discrimination task where either measurement is used according to some partial
intermediate information. The success statistics of the task provides an upper
bound for the amount of incompatibility of the two measurements, as it is
quantified by means of their incompatibility robustness. For a broad class of
unbiased and possibly noisy qubit measurements, one can make this upper bound
coincide with the true value of the robustness by suitably tuning the
preparation of the experiment. We demonstrate this fact in an optical setup,
where the qubit states are encoded into the photons' polarization degrees of
freedom, and incompatibility is directly accessed by virtue of a refined
control on the amplitude, phase and purity of the final projection stage of the
measurements. Our work thus establishes the practical feasibility of a recently
proposed method for the detection of quantum incompatibility.
