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The NUCLEUS experiment aims to measure coherent elastic neutrino nucleus scattering of reactor anti-neutrinos using cryogenic calorimeters. Operating at an overburden of 3 m.w.e., muon-induced backgrounds are expected to be one of the dominant background contributions. Besides a high efficiency to identify muon events passing the experimental setup, the NUCLEUS muon veto has to fulfill tight spatial requirements to fit the constraints given by the experimental site and to minimize the induced detector dead-time. We developed highly efficient and compact muon veto modules based on plastic scintillators equipped with wavelength shifting fibers and silicon photo multipliers to collect and detect the scintillation light. In this paper, we present the full characterization of a prototype module with different light read-out configurations. We conclude that an efficient and compact muon veto system can be built for the NUCLEUS experiment from a cube assembly of the developed modules. Simulations show that an efficiency for muon identification of >99 % and an associated rate of 325 Hz is achievable, matching the requirements of the NUCLEUS experiment.