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Multi-agent LLM debates are increasingly used in domains such as policy, politics, and city planning, where ground truth is often unavailable. Yet existing evaluations rely heavily on outcome-based proxies such as consensus, majority vote, or LLM-as-judge scores, which can miss failures like sycophancy, domination, and premature convergence. We introduce a diagnostic framework that evaluates both debate outcomes and the deliberative process using interpretable metrics for engagement, responsiveness, influence asymmetry, balance, stability, and agent utility. Across real-world debate settings and validation benchmarks, our process-level diagnostics align more closely with human judgments and reveal interaction failures that standard outcome-only measures overlook. These results show that reliable evaluation of multi-agent debates requires measuring not only what answer agents reach, but how they reach it.