Randomized controlled trials (RCTs) are the gold standard for evaluating causal effects but are often costly and difficult to scale; consequently, they are frequently augmented with auxiliary external controls in many applications. Prior approaches for borrowing such data typically rely on exchangeability, under which the external controls are readily usable for inference in the trial population. In practice, however, differences in eligibility criteria, standard of care, and data collection procedures may induce distribution shifts between the RCT and the external controls, rendering exchangeability implausible. In this paper, we propose a novel framework for integrating external controls by explicitly modeling these distribution shifts. We construct augmented estimators by adapting trial-only efficient influence functions through calibration equations that balance the trial and external populations, thereby fully exploiting the external control data even when exchangeability fails. We further develop an adaptive shrinkage estimator that preserves consistency while guaranteeing efficiency dominance over the trial-only benchmark. Synthetic experiments and a real data application demonstrate the practical advantages of the proposed approaches.

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