The present study aims to analyze the physiological and agronomical response to drought among seven local and traditional field-grown Monastrell clones (4, 94, 188, 360, 276, 372, and 373) over four seasons (2018–2021) under optimum irrigation conditions (control) and water stress (stress). We have focussed on measuring Monastrell interclonal variability in plant water relations and leaf gas exchange, vegetative growth, leaf mineral nutrition, yield, water use efficiency (WUE), and grape and wine quality. A classification of the different clones according to drought-tolerance degree and agronomical/oenological performance was established. The classification revealed that (a) The most drought-tolerant clone (i.e., clone 4) was neither the most productive (8600 kg ha−1), the most efficient in terms of water use (average of 25 kg m−3), nor the one that presented a better grape quality; (b) The most productive and efficient clone (i.e., clone 94) (11,566 kg ha−1, average of 30 kg m−3) was also a drought-tolerant clone, but it provided the worst berry and wine qualities with the lowest aromatic/nutraceutical potential, and it is not recommended for premium red wine production; (c) Conversely, clone 360 provided the highest berry quality, but at the expense of a greatly reduced vigor and yield (4000 kg ha−1) and a lower WUEyield (average of 10 kg m−3); (d) Low-vigor clones 372 and 276 were the most sensitive to drought conditions and put more water conservation mechanisms into play, i.e., a tighter control of vine water use and reduced leaf transpiratory surface, under soil water deficit and high vapor pressure deficit (VPD). In addition, these clones reached a balance between drought tolerance, productive water use efficiency (WUEyield), and berry and wine quality, because they provided moderate yields (7400–7700 kg ha−1), a high WUEyield (average between 17–19 kg m−3 applied water), and an enhanced berry and wine quality with greater oenological, nutraceutical, and aromatic potential; (e) High-vigor clone 188 also displayed several mechanisms of drought tolerance (tighter stomatal control of water use), maintained a higher yield (10,500 kg ha−1) and a very high WUEyield (29 kg m−3), enhanced berry quality (similarly to 276 or 372), and improved oenological/aromatic potential, and can also be recommended for the application of low water volume deficit irrigation (DI) strategies under semi-arid conditions.