We establish a necessary and sufficient condition for a Poisson suspension to be prime. The proof is based on the Fock space structure of the $L^{2}$-space of the Poisson suspension. We give examples of explicit infinite measure preserving systems that give way to prime Poisson suspensions, in particular non-singular compact group rotations. We also compare those prime Poisson suspensions to the existing prime transformations known so far, showing our examples are new.