  We determine the quantum phase diagram of the antiferromagnetic spin-1/2 XXZ
model on the triangular lattice as a function of magnetic field and anisotropic
coupling $J_z$. Using the density matrix renormalization group (DMRG) algorithm
in two dimensions we establish the locations of the phase boundaries between a
plateau phase with 1/3 N\'eel order and two distinct coplanar phases. The two
coplanar phases are characterized by a simultaneous breaking of both
translational and U(1) symmetries, which is reminiscent of supersolidity. A
translationally invariant umbrella phase is entered via a first order phase
transition at relatively small values of $J_z$ compared to the corresponding
case of ferromagnetic hopping and the classical model. The phase transition
lines meet at two tricritical points on the tip of the lobe of the plateau
state, so that the two coplanar states are completely disconnected.
Interestingly, the phase transition between the plateau state and the upper
coplanar state changes from second order to first order for large values of
$J_z > 2.5J$.
