The ability of core-shell nanowires to overcome existing limitations of heterostructures is one of the key ingredients for the design of next generation devices. This requires a detailed understanding of the mechanism for strain relaxation in these systems in order to eliminate strain-induced defect formation and thus to boost important electronic properties such as carrier mobility. Here we demonstrate how the hole mobility of -oriented Ge-Si core-shell nanowires can be substantially enhanced thanks to the realization of large band offset and coherent strain in the system, reaching values as high as 4200 cm2/(Vs) at 4 K and 1600 cm2/(Vs) at room temperature for high hole densities of 1019 cm-3. We present a direct correlation of (i) mobility, (ii) crystal direction, (iii) diameter, and (iv) coherent strain, all of which are extracted in our work for individual nanowires. Our results imply -oriented Ge-Si core-shell nanowires as a promising candidate for future electronic and quantum transport devices.