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We describe a new instrument, the Argonne Auger Radioisotope Microscope (ARM), capable of characterizing the Auger electron emission of radionuclides, including candidates relevant in nuclear medicine. Our approach relies on event-by-event ion-electron coincidence, time-of-flight, and spatial readout measurement to determine correlated electron multiplicity and energy distributions of Auger decays. We present a proof-of-principle measurement with the ARM using X-ray photoionization of stable krypton beyond the K-edge and identify a bifurcation in the electron multiplicity distribution depending on the emission of K-LX electrons. Extension of the ARM to the characterization of radioactive sources of Auger electron emissions is enabled by the combination of two recent developments: (1) cryogenic buffer gas beam technology to introduce Auger emitters into the detection region with well-defined initial conditions, and (2) large-area micro-channel plate detectors with multi-hit detection capabilities to simultaneously detect multiple electrons emitted in a single decay. The ARM will generate new experimental data on Auger multiplicities that can be used to benchmark atomic relaxation and decay models. This data will provide insight into the low-energy regime of Auger electrons where intensity calculations are most challenging and experimental data is limited. In particular, accurate multiplicity data of the low-energy regime can be used to inform oncological dosimetry models, where electron energies less than 500 eV are known to be most effective in damaging DNA and cell membranes.