In this work, we analyze Lagrangian material coherence in dynamic open domains. We derive and apply new theory and efficient schemes to extract material sets in dynamic flow fields that remain the most or the least coherent throughout the time interval of interest, with special attention to realistic ocean domains that have multiple time-dependent inlets and/or outlets. First, the partial differential equation (PDE)-based method of composition is extended to efficiently compute flow maps in open domains, evolving a dynamic mask field without compounding numerical errors. This permits the use of existing grid based PDE solvers to compute flow maps on their dynamic non-regular domain. Inherent parallelization capabilities with accuracy as trajectory-based schemes but importantly with also an optimal grid-based resolution make this method very attractive. Second, we derive a novel approach to compute material sets in dynamic fluid flows that undergo minimal stress throughout the considered time interval. The level sets of the proposed metric, called the ‘extended polar distance’, yield material subdomains that remain rigid (i.e. only undergo translation and rotation) throughout the time interval of interest up to a certain tolerance. This metric and the corresponding persistently coherent sets and incoherent sets are computed using the PDE-based flow map computation. We further relate the extended polar distance and the diffusion barrier strength metric and show that the extended polar distance rigorously cumulates the tendency of a material subdomain to be prone to diffusion and the average strain it undergoes. We utilize the new theory and numerical methods to analyze Lagrangian coherence in analytical and realistic scenarios – an analytical unsteady double gyre flow and a realistic simulation in the Southern Pacific Ocean. The former helps us better understand the proposed theory in practice, and highlights the evolution of coherent, persistently coherent, and incoherent sets. In the latter Southern Pacific Ocean application, we find that the surface regions around Palau island are highly incoherent due to the steep topography and complex interactive dynamics. However, we also find a rigid set advected by the larger-scale currents around the Island, retrieving its shape at the end, as well as a persistently rigid set that approximately maintains its shape throughout the time interval, maximally resisting advective stretching and diffusive transport.