Quantum fluctuations in the effective spin-12 layered triangular-lattice quantum Heisenberg antiferromagnet Ba3CoSb2O9 lift the classical degeneracy of the antiferromagnetic ground state in magnetic field, producing a series of novel spin structures for magnetic fields applied within the crystallographic ab plane, including a celebrated collinear “up-up-down” spin ordering with magnetization equal to 1/3 of the saturation magnetization over an extended field range. Theoretically unresolved, however, are the effects of interlayer antiferromagnetic coupling and transverse magnetic fields on the ground states of this system. Additional magnetic field induced phase transitions are theoretically expected and in some cases have been experimentally observed, but details regarding their number, location, and physical character appear inconsistent with the predictions of existing models. Conversely, an absence of experimental measurements as a function of magnetic-field orientation has left other key predictions of these models untested. To address these issues, we have used specific heat, neutron diffraction, thermal conductivity, and magnetic torque measurements to map out the phase diagram as a function of magnetic field intensity and orientation relative to the crystallographic ab plane. For H||ab, we have discovered an additional magnetic field induced phase transition at low temperature and an unexpected tetracritical point in the high-field phase diagram, which coupled with the apparent second-order nature of the phase transitions eliminates several theoretically proposed spin structures for the high-field phases. Our calorimetric measurements as a function of magnetic field orientation are in general agreement with theory for field-orientation angles close to plane parallel (H||a) but diverge at angles near plane perpendicular; a predicted convergence of two phase boundaries at finite angle and a corresponding change in the order of the field induced phase transition are not observed experimentally. Our results emphasize the role of interlayer coupling in selecting and stabilizing field induced phases, provide guidance on the nature of the magnetic order in each phase, and reveal the need for new physics to account for the nature of magnetic ordering in this archetypal two-dimensional spin-12 triangular-lattice quantum Heisenberg antiferromagnet.