We perform self-consistent Schr\"odinger-Poisson calculations with exchange and correlation corrections to determine the electron/hole gas in a radial hetero-junction formed in a modulation doped GaAs/AlGaAs core-multi-shell nanowire (CSNW) which is n-/p-doped. Realistic composition and geometry are mapped on an symmetry compliant two-dimensional grid, and the inversion/accumulation layers are obtained assuming mid-gap Fermi energy pinning at the surface. We show that the electron and hole gases can be tuned to different localizations and symmetries inside the core as a function of the doping level. Contrary to planar hetero-junctions, conduction electrons do not form a uniform 2D electron gas (2DEG) localized at the GaAs/AlGaAs interface, but rather show a transition between i) an isotropic, cylindrical distribution deep in the GaAs core (low doping), and ii) a set of six tunnel-coupled quasi-1D channels at the edges of the interface (high doping). Holes, on the other hand, are much more localized at the GaAs/AlGaAs interface and form either i) six separated planar 2DEGs at the GaAs/AlGaAs interfaces (low doping), ii) a quasi uniform six-fold bent 2DEG (intermediate doping), or iii) six tunnel-coupled quasi-1D channels at the edges (high doping). We also simulate the electron/hole gas in a CSNW-based field effect transistor. The field generated by a back-gate may easily deform the electron or hole gas, breaking the six-fold symmetry. Single 2DEGs at one interface or multiple quasi-1D channels are shown to form as a function of voltage intensity, polarity, and carrier type.