Structured illumination microscopy (SIM) is a powerful super-resolution imaging technique that uses patterned illumination to down-modulate high spatial-frequency information of samples. However, the presence of spatially-dependent aberrations can severely disrupt the illumination pattern, limiting the quality of SIM imaging. Conventional adaptive optics (AO) techniques that employ wavefront correctors at the pupil plane are not capable of effectively correcting these spatially-dependent aberrations. We introduce the Tandem Aberration Correction Optics (TACO) approach that combines both pupil AO and conjugate AO for aberration correction in SIM. TACO incorporates a deformable mirror (DM) for pupil AO in the detection path to correct for global aberrations, while a spatial light modulator (SLM) is placed at the plane conjugate to the aberration source near the sample plane, termed conjugate AO, to compensate spatially-varying aberrations in the illumination path. Our numerical simulations and experimental results show that the TACO approach can recover the illumination pattern close to an ideal condition, even when severely misshaped by aberrations, resulting in high-quality super-resolution SIM reconstruction. The TACO approach resolves a critical traditional shortcoming of aberration correction for structured illumination. This advance significantly expands the application of SIM imaging in the study of complex, particularly biological, samples and should be effective in other wide-field microscopies.