In cross-cultural recipe adaptation, the goal is not only to ensure cultural appropriateness and retain the original dish's essence, but also to provide diverse options for various dietary needs and preferences. Retrieval Augmented Generation (RAG) is a promising approach, combining the retrieval of real recipes from the target cuisine for cultural adaptability with large language models (LLMs) for relevance. However, it remains unclear whether RAG can generate diverse adaptation results. Our analysis shows that RAG tends to overly rely on a limited portion of the context across generations, failing to produce diverse outputs even when provided with varied contextual inputs. This reveals a key limitation of RAG in creative tasks with multiple valid answers: it fails to leverage contextual diversity for generating varied responses. To address this issue, we propose CARRIAGE, a plug-and-play RAG framework for cross-cultural recipe adaptation that enhances diversity in both retrieval and context organization. To our knowledge, this is the first RAG framework that explicitly aims to generate highly diverse outputs to accommodate multiple user preferences. Our experiments show that CARRIAGE achieves Pareto efficiency in terms of diversity and quality of recipe adaptation compared to closed-book LLMs.