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We characterize a high-density sample of negatively charged silicon-vacancy (SiV$^-$) centers in diamond using collinear optical multidimensional coherent spectroscopy. By comparing the results of complementary signal detection schemes, we identify a hidden population of \ce{SiV^-} centers that is not typically observed in photoluminescence, and which exhibits significant spectral inhomogeneity and extended electronic $T_2$ times. The phenomenon is likely caused by strain, indicating a potential mechanism for controlling electric coherence in color-center-based quantum devices.