We address key challenges in Dataset Aggregation (DAgger) for real-world contact-rich manipulation: how to collect informative human correction data and how to effectively update policies with this new data. We introduce Compliant Residual DAgger (CR-DAgger), which contains two novel components: 1) a Compliant Intervention Interface that leverages compliance control, allowing humans to provide gentle, accurate delta action corrections without interrupting the ongoing robot policy execution; and 2) a Compliant Residual Policy formulation that learns from human corrections while incorporating force feedback and force control. Our system significantly enhances performance on precise contact-rich manipulation tasks using minimal correction data, improving base policy success rates by 64% on four challenging tasks (book flipping, belt assembly, cable routing, and gear insertion) while outperforming both retraining-from-scratch and finetuning approaches. Through extensive real-world experiments, we provide practical guidance for implementing effective DAgger in real-world robot learning tasks. Result videos are available at: https://compliant-residual-dagger.github.io