Cultural image retrieval is essential for organizing and analyzing large-scale digital heritage resources. However, existing multimodal retrieval methods usually encode a query as a holistic embedding, making it difficult to preserve logical operators and cultural relations during retrieval. Therefore, we introduce Multimodal Boolean Logic Culture Retrieval, a novel task that moves beyond general visual similarity by enforcing strict Boolean constraints over structured cultural elements. To tackle this, we propose *CLAIR*, a training-free framework that reformulates the retrieval objective to seamlessly integrate logical operations directly within the multimodal embedding space. Furthermore, we construct CulTi2.0, a dedicated benchmark designed to evaluate diverse logical query forms based on cultural visual atoms. Extensive experiments show that while current multimodal models struggle with explicit logical queries, *CLAIR* provides consistent and generalizable improvements as a plug-and-play solution across various datasets and model families. Code and dataset are publicly available at https://github.com/AI4Culture/CulTi2.0.

Cross-modal retrieval is essential for interpreting cultural heritage data, but its effectiveness is often limited by incomplete or inconsistent textual descriptions, caused by historical data loss and the high cost of expert annotation. While large language models (LLMs) offer a promising solution by enriching textual descriptions, their outputs frequently suffer from hallucinations or miss visually grounded details. To address these challenges, we propose C^3, a data augmentation framework that enhances cross-modal retrieval performance by improving the completeness and consistency of LLM-generated descriptions. C^3 introduces a bidirectional validation mechanism to assess semantic coverage using both visual cues and language-model outputs. Furthermore, to mitigate factual inconsistencies, we formulate a Markov Decision Process to supervise Chain-of-Thought reasoning, guiding consistency verification through adaptive query control. Experiments on the cultural heritage dataset CulTi and general benchmarks MSCOCO and Flickr30K demonstrate that C^3 achieves state-of-the-art performance in both fine-tuned and zero-shot settings.