We continue the study of adapted optimal transport in the Gaussian discrete-time setting. We introduce a space of filtered Gaussian processes, where the randomness and the flow of information are driven by a Gaussian white noise process. On this space, we derive explicitly the adapted 2-Wasserstein distance AW_2, and establish various results including metric completeness and geodesic convexity. We also prove that this space is the AW_2-completion of the space of Gaussian distributions. In particular, the adapted 2-Wasserstein distance can be expressed in terms of a constrained Procrustes problem between the Cholesky factors. Next, we study the adapted Brenier coupling which is the multivariate analogue of the Knothe-Rosenblatt coupling and can be viewed as a myopic solution to the adapted transport problem. We compare its transport cost with the adapted 2-Wasserstein distance in a random matrix setting. Finally, we show that Gelbrich's lower bound of the 2-Wasserstein distance does not generally hold in the adapted setting, and provide a martingale difference condition under which it holds. (Joint work with Madhu Gunasingam.)