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Quantile mapping (QM) is a technique often used for statistical post-processing (SPP) of climate model simulations, in order to adjust their biases relative to a selected reference product and/or to downscale their resolution. However, when QM is applied in univariate mode, there is a risk of generating other problems, like intervariable physical inconsistency (PI). Here, such a risk is investigated with daily temperature minimum (T-min) and maximum (T-max), for which the relationship T-min > T-max would be inconsistent with the definition of the variables. QM is applied to an ensemble of 78 daily CMIP5 simulations over Hudson Bay for the application period 1979-2100, with Climate Forecast System Reanalysis (CFSR) selected as the reference product during the calibration period 1979-2010. This study's specific objectives are as follows: to investigate the conditions under which PI situations are generated; to test whether PI may be prevented simply by tuning some of the QM technique's numerical choices; and to compare the suitability of alternative approaches that hinder PI by design. Primary results suggest that PI situations appear preferentially for small values of the initial (simulated) diurnal temperature range (DTR), but the differential between the respective biases of T-min and T-max also plays an important role; one cannot completely prevent the generation of PI simply by adjusting QM parameters and options, but forcing preservation of the simulated long-term trends generates fewer PI situations; for avoiding PI between T-min and T-max, the present study supports a previous recommendation to directly post-process T-max and DTR before deducing T-min.