  We investigate the phase diagram of the anisotropic spin-1/2 triangular
lattice antiferromagnet, with interchain diagonal exchange J' much weaker than
the intrachain exchange J. We find that fluctuations lead to a competition
between (commensurate) collinear antiferromagnetic and (zig-zag) dimer orders.
Both states differ in symmetry from the spiral order known to occur for larger
J', and are therefore separated by quantum phase transitions from it. The
zero-field collinear antiferromagnet is succeeded in a magnetic field by
magnetically-ordered spin-density-wave and cone phases, before reaching the
fully polarized state. Implications for the anisotropic triangular magnet
Cs_2CuCl_4 are discussed.
