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Micron-sized superconducting interference devices ($\mu$-SQUIDs) based onconstrictions optimized for minimizing thermal runaway are shown to exhibitvoltage oscillations with applied magnetic flux despite their hystereticbehavior. We explain this remarkable feature by a significant supercurrentcontribution surviving deep into the resistive state, due to efficient heatevacuation. A resistively shunted junction model, complemented by a thermalbalance determining the amplitude of the critical current, describes well allexperimental observations, including the flux modulation of the (dynamic)retrapping current and voltage by introducing a single dimensionless parameter.Thus hysteretic $\mu$-SQUIDs can be operated in the voltage read-out mode witha faster response. The quantitative modeling of this regime incorporating bothheating and phase dynamics paves the way for further optimization of$\mu$-SQUIDs for nano-magnetism.

Josephson Coupling in the Dissipative State of a Thermally Hysteretic $\mu$-SQUID