Imitation learning (IL) is a powerful approach for acquiring optimal policies from demonstrated behaviors. However, applying IL to a large group of agents is arduous due to the exponential surge in interactions with an increase in population size. Mean Field Theory provides an efficient tool for analyzing multi-agent problems by gathering information at the population level. Although the approximation is tractable, restoring mean field Nash equilibria (MFNE) from demonstrations is challenging. Furthermore, many real-world problems, including traffic network equilibrium induced by public routing recommendations and pricing equilibrium of goods on E-commerce platforms, cannot be explained by the classic MFNE concept. In both cases, the intervention of the platform introduces correlation devices to the equilibrium. To address this issue, we propose a novel solution concept called Adaptive Mean Field Correlated Equilibrium (AMFCE) that generalizes MFNE. We establish a framework based on IL and AMFCE that recovers the AMFCE policy from real-world demonstrations. Our framework characterizes mean-field evolution using signatures from the rough path theory, and it has the significant benefit of recovering both the equilibrium policy and correlation device from data. We test our framework against state-of-the-art IL algorithms for mean field games (MFGs) on several tasks, including a real-world traffic flow prediction problem. Our results demonstrate the effectiveness of our proposed method and its potential for predicting and explaining large population behavior under correlated signals.