We study identifiability of linear Andersson-Madigan-Perlman (AMP) chain graph models, which are a common generalization of linear structural equation models and Gaussian graphical models. AMP models are described by DAGs on chain components which themselves are undirected graphs. For a known chain component decomposition, we show that the DAG on the chain components is identifiable if the determinants of the residual covariance matrices of the chain components are equal (or more generally, monotone non-decreasing in topological order). This condition extends the equal variance identifiability criterion for Bayes nets, and it can be generalized from determinants to any super-additive function on positive semidefinite matrices. When the component decomposition is unknown, we describe conditions that allow recovery of the full structure using a polynomial time algorithm based on submodular function minimization. We also conduct experiments comparing our algorithm's performance against existing baselines.