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Many tasks, particularly those involving interaction with the environment, are characterized by high variability, making robotic autonomy difficult. One flexible solution is to introduce the input of a human with superior experience and cognitive abilities as part of a shared autonomy policy. However, current methods for shared autonomy are not designed to address the wide range of necessary corrections (e.g., positions, forces, execution rate, etc.) that the user may need to provide to address task variability. In this paper, we present corrective shared autonomy, where users provide corrections to key robot state variables on top of an otherwise autonomous task model. We provide an instantiation of this shared autonomy paradigm and demonstrate its viability and benefits such as low user effort and physical demand via a system-level user study on three tasks involving variability situated in aircraft manufacturing.