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Multiple scattering is a major barrier that limits the optical imaging depth in scattering media. In order to alleviate this effect, we demonstrate aberration-diverse optical coherence tomography (AD-OCT), which exploits the phase correlation between the deterministic signals from single-scattered photons to suppress the random background caused by multiple scattering and speckle. AD-OCT illuminates the sample volume with diverse aberrated point spread functions, and computationally removes these intentionally applied aberrations. After accumulating 12 astigmatism-diverse OCT volumes, we show a 10 dB enhancement in signal-to-background ratio via a coherent average of reconstructed signals from a USAF target located 7.2 scattering mean free paths below a thick scattering layer, and a 3× speckle contrast reduction from an incoherent average of reconstructed signals inside the scattering layer. This AD-OCT method, when implemented using astigmatic illumination, is a promising approach for ultra-deep volumetric optical coherence microscopy.