Interacting flows are found in a range of aviation-relevant relevant technologies, including flow control devices, engine combustors and augmentors, aero-optics flow interactions, and aerodynamic control surfaces. A somewhat limited literature on interacting jets and wakes indicates that the structure and dynamics of these flowfields, including both large-scale coherent dynamics stemming from hydrodynamic instability and turbulent fluctuations, is fundamentally different from that of the single jet or single wake flowfield. In particular, the interacting flowfields experience variations in vortex shedding frequency and phase that change as the distance between the adjacent jets or wakes is varied. The goal of this work is to understand large-scale, intermittent dynamics of turbulent interacting wakes and jets using an improved reduced-order modeling strategy, cluster-based reduced-order modeling, to capture these dynamics. We compare the dynamics of a three-wake system at two spacings to that of a single wake flowfield using both proper orthogonal decomposition (POD) as well as the cluster-based method (CROM). The CROM is able to capture the expected dynamics of the single wake, and the results are analogous to those from POD. However, CROM reveals a much more complicated set of dynamics in the interacting wake cases, including the existence of two sets of dynamics that intermittently appear, that the POD was unable to detect. CROM is used to quantify these dynamics and understand the effect of bluff-body spacing on the three-wake flowfields.