How many processes have Poisson counts?

The Poisson process has the well-known Poisson count property: the count of points in any subset of the carrier space has a Poisson distribution. To specify the complete distribution of a point process it is necessary and sufficient to specify all of the joint distributions of the counts of points in any (finite disjoint) collection of bounded sets in the carrier space. Suppose that only the Poisson count property is specified for a random collection of points. We reveal the circumstances in which the Poisson count property does indeed determine the distribution. Curiously, there is a 'phase transition' in this property with the boundary being mean measures having 2 atoms.