Forsström et al. [8] recently introduced a large class of $\{0,1\}$-valued processes that they named Poisson representable. In addition to deriving several interesting properties for these processes, their main focus was determining which processes are contained in this class. In this paper, we derive new characteristics for Poisson representable processes in terms of certain mixing properties. Using these, we argue that neither the upper invariant measure of the supercritical contact process on $\mathbb{Z}^d$ nor the plus state of the Ising model on $\mathbb{Z}^2$ within the phase transition regime is Poisson representable. Moreover, we show that on $\mathbb{Z}^d$, $d\geq 2$, any non-extremal translation invariant state of the Ising model cannot be Poisson representable. Together, these results provide answers to questions raised in [8].