Real-world reinforcement learning has enabled contact-rich manipulation tasks to be trained directly on physical robots, but unsafe interactions during online learning remain a key barrier to practical deployment. We propose Safe Evidential Real-World Reinforcement Learning (SEVI), a framework for reliability-aware safety intervention during real-world policy learning. SEVI introduces an Evidential Safety Critic that estimates both future constraint-violation risk and epistemic uncertainty in a single forward pass. During online learning, SEVI evaluates each task-policy action before execution; if the action is predicted to be unsafe or the prediction is insufficiently supported by evidence, SEVI overrides it with an offline-trained recovery policy that guides the robot toward a lower-risk region. We evaluate SEVI on two simulated manipulation tasks and a real-world USB insertion task under in-distribution and distribution-shifted conditions. Across task-learning and safety-critic baselines, SEVI achieves the strongest trajectory-level safety prediction, the highest final task success, and the lowest unsafe-event counts. SEVI also reduces unsafe events by 13.4 to 57.1 percent relative to recovery-based safe RL while preserving low-latency execution-time filtering. The project page is available at: https://sevi-safe.github.io/

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