The type of object I mean the process to yield is, I think, a 2d distribution in the sense of schwartz, in that it's the kind of thing that you can query by throwing, say, a rectangle in R2 at it, and it'll tell you what answer you get if you integrate over that rectangle.
The defining properties of the distribution are these:
(a) For any rectangle R with area A, the probability that integral over R has a particular value x is
i.e. it's distributed ~ Gamma(A, 1), and
(b) the distribution of disjoint rectangles is independent.
If this is a reasonable thing, I'm tempted to convolve it with a kernel that's nice and compactly supported and C∞ and arrive a nice random smooth function from R2 to R≥0 that has short-range but no long-range correlations.