We combine the dynamics of open quantum systems with interferometry and interference, introducing the concept of an open system interferometer. By considering a single photon in a Mach-Zehnder interferometer, where the polarization (open system) and frequency (environment) of the photon interact, we theoretically show that inside the interferometer, pathwise polarization dephasing dynamics is Markovian while the joint dynamics displays non-Markovian features. Outside the interferometer and due to interference, the open system displays rich dynamical features with distinct alternatives: only one path displays non-Markovian memory effects, both paths individually display them, or no memory effects appear at all. The scheme allows one to (1) probe the optical path difference inside the interferometer by studying pathwise non-Markovianity outside the interferometer, and (2) introduce pathwise dissipative features for the open system dynamics even though the system-environment interaction itself contains only dephasing. Due to the path dependencies, our results are tightly connected to quantum erasure. In general, our results open up alternative ways to control open system dynamics and for fundamental studies of quantum physics. © 2021 American Physical Society.