Large Language Models (LLMs) often hallucinate, limiting their reliability in sensitive applications. In black-box settings, several self-consistency-based techniques have been proposed for hallucination detection. We empirically show that these methods perform nearly as well as a supervised (black-box) oracle, leaving limited room for further gains within this paradigm. To address this limitation, we explore cross-model consistency checking between the target model and an additional verifier LLM. With this extra information, we observe improved oracle performance compared to purely self-consistency-based methods. We then propose a budget-friendly, two-stage detection algorithm that calls the verifier model only for a subset of cases. It dynamically switches between self-consistency and cross-consistency based on an uncertainty interval of the self-consistency classifier. We provide a geometric interpretation of consistency-based hallucination detection methods through the lens of kernel mean embeddings, offering deeper theoretical insights. Extensive experiments on QA-style hallucination detection benchmarks show that this approach maintains high detection performance while significantly reducing computational cost.

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