Controlling the crystallographic phase purity of III-V nanowires is notoriously difficult, yet this is essential for future nanowire devices. Reported methods for controlling nanowire phase require dopant addition, or a restricted choice of nanowire diameter, and only rarely yield a pure phase. Here we demonstrate that phase-perfect nanowires, of arbitrary diameter, can be achieved simply by tailoring basic growth parameters: temperature and V/III ratio. Phase purity is achieved without sacrificing important specifications of diameter and dopant levels. Pure zinc blende nanowires, free of twin defects, were achieved using a low growth temperature coupled with a high V/III ratio. Conversely, a high growth temperature coupled with a low V/III ratio produced pure wurtzite nanowires free of stacking faults. We present a comprehensive nucleation model to explain the formation of these markedly different crystal phases under these growth conditions. Critical to achieving phase purity are changes in surface energy of the nanowire side facets, which in turn are controlled by the basic growth parameters of temperature and V/III ratio. This ability to tune crystal structure between twin-free zinc blende and stacking-fault-free wurtzite not only will enhance the performance of nanowire devices but also opens new possibilities for engineering nanowire devices, without restrictions on nanowire diameters or doping.