Gaussian double Markovian models consist of covariance matrices with specified zeros and specified zeros in their inverses. Geometrically, these are intersections of graphical models with inverse graphical models. We describe the semi-algebraic geometry of these models, in particular their dimension, smoothness, connectedness, and vanishing ideals. We give a geometric proof of an exponential family heuristic for a smoothness criterion of Zwiernik. We also continue investigations of singular loci initiated by Drton and Xiao. This is joint work with Tobias Boege, Andreas Kretschmer, and Frank Röttger