  Two classes of interacting particle systems on $\mathbb{Z}$ are shown to be
Pfaffian point processes at fixed times, and for all deterministic initial
conditions. The first comprises coalescing and branching random walks, the
second annihilating random walks with pairwise immigration. Various limiting
Pfaffian point processes on $\mathbb{R}$ are found by diffusive rescaling,
including the point set process for the Brownian net.
