Face Anti-Spoofing (FAS) must distinguish genuine faces from presentation attacks. A unified Multimodal Large Language Model (MLLM) is an attractive solution: it detects physical attacks (print, replay, mask) in one model and produces interpretable forensic evidence via Chain-of-Thought (CoT). A central obstacle to post-training such a model with reinforcement learning is that FAS reasoning has no rule-verifiable ground truth: only the binary verdict is checkable, while the reasoning chain cannot be verified by any hand-crafted rule. Optimizing the verdict alone is therefore a sparse, hackable signal. The model can reach the correct answer through a shortcut such as a false-rejection bias, which does not generalize. We address this with a grounding-aware judge reward. A 32B vision-language judge scores each reasoning chain against the image on evidence anchoring, logical coherence, and domain knowledge, turning the image itself into the verification signal. The reward is applied in a Group Relative Policy Optimization (GRPO) stage after CoT supervised fine-tuning. It is the judge, rather than the RL objective, that supplies the process supervision a verdict-only reward cannot. Our study starts from a model that already performs well In-Domain (ID) after CoT-based supervised fine-tuning and asks how to improve its Out-of-Domain (OOD) generalization. We find that grounding-aware rewards are an effective means to that end. We validate the method on OOD generalization, where the reward reduces ACER on two physical benchmarks (LCC-FASD 0.288 to 0.273, MSU-MFSD 0.431 to 0.409) while preserving ID accuracy (CelebA-Spoof 0.033 to 0.035). Against ResNet-50/101 and CLIP-ViT-B/16 baselines, the MLLM dominates ID and matches or surpasses the strongest baseline OOD. To our knowledge this is the first use of LLM-as-judge process supervision for FAS.

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