For a sequence $X_1,X_2,X_3,\ldots$ of random variables, what it means to say $X_1$ is correlated with $X_2$ is unambiguous. It may be that the bigger $X_1$ is, the bigger $X_2$ is likely to be. If, as in conventional Kolmogorovian probability theory, we regard all these random variables as functions on a probability space, it makes sense to speak of different realizations $X_1(\omega_1), X_2(\omega_1), X_3(\omega_1),\ldots$ and $X_1(\omega_2), X_2(\omega_2), X_3(\omega_2),\ldots$. I.e. run the random process once; get one set of values of $X_1,X_2,X_3,\ldots$; run it again, get another, etc.
But intuitively, it would make sense to say that the size and shape of one Voronoi cell in something like a Poisson process in the plane are correlated with those of its neighbor. But if we run the process once and get neighboring Voronoi cells $X_1$ and $X_2$, and run the process again, then which cells in the second realization correspond to $X_1$ and $X_2$ in the first one? There seems to be no reasonable answer. A Voronoi cell has no immortal soul, a coin toss does. The first thing I think of is to let $X_1$ be the Voronoi cell that contains the origin. But then the expected value of its size is not the same as expected size of an arbitray Voronoi cell (it's bigger!), and which of its neighbors could be $X_2$? The very number of its neighbors varies from one $\omega$ to the next.
Is there some good way to rescue a concept of correlation of size and shape of neighboring Voronoi cells?