  Symmetry breaking is reported for continuous families of solitons in the
nonlinear Schr\"odinger equation with a two-dimensional complex potential. This
symmetry-breaking bifurcation is forbidden in generic complex potentials.
However, for a special class of partially parity-time-symmetric potentials,
such symmetry breaking is allowed. At the bifurcation point, two branches of
asymmetric solitons bifurcate out from the base branch of symmetry-unbroken
solitons. Stability of these solitons near the bifurcation point are also
studied, and two novel stability properties for the bifurcated asymmetric
solitons are revealed. One is that at the bifurcation point, zero and simple
imaginary linear-stability eigenvalues of asymmetric solitons can move directly
into the complex plane and create oscillatory instability. The other is that
the two bifurcated asymmetric solitons, even though having identical powers and
being related to each other by spatial mirror reflection, can possess different
types of unstable eigenvalues and thus exhibit non-reciprocal nonlinear
evolutions under random-noise perturbations.
