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.