  We consider an S=1 Heisenberg chain with strong exchange (Delta) and
single--ion uniaxial anisotropy (D) in a magnetic field (B) along the symmetry
axis. The low energy spectrum is described by an effective S=1/2 XXZ model that
acts on two different low energy sectors for a given window of fields. The
vacuum of each sector exhibits Ising-like antiferromagnetic ordering that
coexists with the finite spin stiffness obtained from the exact solution of the
effective XXZ model. In this way, we demonstrate the existence of a spin
supersolid phase. We also compute the full Delta-B quantum phase diagram by
means of a quantum Monte Carlo simulation.
