Causal disentanglement aims to uncover a representation of data using latent variables that are interrelated via a causal model. Such a representation is identifiable if the latent model that explains the data is unique. In this work, we focus on the scenario where observational and interventional data are available, with each intervention changing the mechanism of a latent variable. When the causal variables are fully observed, statistically consistent algorithms have been developed to identify the causal model under faithfulness assumptions. We here show that identifiability can still be achieved with unobserved causal variables, given a generalized notion of faithfulness. Our results guarantee that we can recover the latent causal model up to an equivalence class and predict the effect of unseen combinations of interventions, in the limit of infinite data.