Analytic shrinkage is a statistical technique that offers a fast alternative to cross-validation for the regularization of covariance matrices and has appealing consistency properties. We show that the proof of consistency implies bounds on the growth rates of eigenvalues and their dispersion, which are often violated in data. We prove consistency under assumptions which do not restrict the covariance structure and therefore better match real world data. In addition, we propose an extension of analytic shrinkage --orthogonal complement shrinkage-- which adapts to the covariance structure. Finally we demonstrate the superior performance of our novel approach on data from the domains of finance, spoken letter and optical character recognition, and neuroscience.