We study statistical models of regular Gaussian distributions given by assumptions about the signs of partial correlations. This includes conditional independence models and graphical modeling devices such as Markov and Bayes networks. For these models, we consider the following basic questions: (1) How hard is it (complexity-theoretically) to check if the model specification is inconsistent? (2) If it is consistent, how hard is it (algebraically) to write down a covariance matrix from the model? (3) How badly shaped (homotopy-theoretically) can these models be? For all of these questions the answer is "it is as bad as it could possibly be".