\textbf{RE}trieval-\textbf{A}ugmented \textbf{L}LM-based \textbf{M}achine \textbf{T}ranslation (REAL-MT) shows promise for knowledge-intensive tasks like idiomatic translation, but its reliability under noisy retrieval contexts remains poorly understood despite this being a common challenge in real-world deployment. To address this gap, we propose a noise synthesis framework and new metrics to evaluate the robustness of REAL-MT systematically. Using this framework, we instantiate REAL-MT with Qwen-series models, including standard LLMs and large reasoning models (LRMs) with enhanced reasoning, and evaluate their performance on idiomatic translation across high-, medium-, and low-resource language pairs under synthesized noise. Our results show that low-resource language pairs, which rely more heavily on retrieved context, degrade more severely under noise than high-resource ones and often produce nonsensical translations. Although LRMs possess enhanced reasoning capabilities, they show no improvement in error correction and are even more susceptible to noise, tending to rationalize incorrect contexts. We find that this stems from an attention shift away from the source idiom to noisy content, while confidence increases despite declining accuracy, indicating poor calibration. To mitigate these issues, we investigate training-free and fine-tuning strategies, which improve robustness at the cost of performance in clean contexts, revealing a fundamental trade-off. Our findings highlight the limitations of current approaches, underscoring the need for self-verifying integration mechanisms.