In a random Gaussian wave field there is on average one vortex (phase singularity) for every two coherence areas. These vortices exhibit an unexpected, highly surprising correlation-nearest neighbors are 90% anticorrelated in sign. We show that on the zero crossings of the real or imaginary parts of the field adjacent vortices must be of opposite sign. This principle, which is unaffected by boundaries, accounts for the observed sign anticorrelations. We also show how the sign of any single vortex in a random wave field determines the sign of all other vortices in the wave field.