Retrieving real-time information is a fundamental capability for search-integrated agents in real-world applications. However, existing benchmarks are predominantly static and therefore fail to capture the temporal dynamics of information and the continuously evolving nature of real-world knowledge. To address this limitation, we propose RT-QA, a dynamic evaluation framework that leverages executable code workflows to retrieve up-to-date answers at evaluation time. Specifically, we construct an agent-driven pipeline that autonomously generates code for web crawling and DOM-based answer extraction to produce real-time ground truth. To ensure robust evaluation over time, the pipeline further incorporates a self-repair mechanism to adapt to changes in web page structures. RT-QA spans 12 domains (e.g., Finance, Sports) with 320 Chinese questions categorized into three difficulty levels. Extensive evaluations of state-of-the-art models (e.g., GPT-5.2, GLM-4.7) reveal significant limitations in real-time adaptability: even the best models achieve only 46% accuracy. Our analysis highlights two primary failure modes: (1) Lazy Retrieval, where agents rely on search snippets instead of deeply scanning specific websites for information (20% of failures); and (2) Temporal Confusion, a cognitive error where agents retrieve a historical date (e.g., an event in 2024) and fail to re-anchor to the current time (2026) for subsequent reasoning. These findings suggest that future agents require not just better retrieval strategies, but robust temporal state management.