  Two overall neutral surfaces with positive and negative charged domains
("patches") have been shown in recent experiments to exhibit long-range
attraction when immersed in an ionic solution. Motivated by the experiments, we
calculate analytically the osmotic pressure between such surfaces within the
Poisson-Boltzmann framework, using a variational principle for the
surface-averaged free energy. The electrostatic potential, calculated beyond
the linear Debye-H\"uckel theory, yields an {\it overall attraction} at large
inter-surface separations, over a wide range of the system's controlled length
scales. In particular, the attraction is stronger and occurs at smaller
separations for surface patches of larger size and charge density. In this
large patch limit, we find that the attraction-repulsion crossover separation
is inversely proportional to the square of the patch charge-density and to the
Debye screening length.
