Starting from limited measurements of a turbulent flow, data assimilation (DA) attempts to estimate all the spatio-temporal scales of motion. Success is dependent on whether the system is observable from the measurements, or how much of the initial turbulent field is encoded in the available measurements. Adjoint-variational DA minimizes the discrepancy between the true and estimated measurements by optimizing the initial velocity or vorticity field (the ‘state space’). Here we propose to instead optimize in a lower-dimensional latent space which is learned by implicit rank minimizing autoencoders. Assimilating in latent space, rather than state space, redefines the observability of the measurements and identifies the physically meaningful perturbation directions which matter most for accurate prediction of the flow evolution. When observing coarse-grained measurements of two-dimensional Kolmogorov flow at moderate Reynolds numbers, the proposed latent-space DA approach estimates the full turbulent state with a relative error improvement of two orders of magnitude over the standard state-space DA approach. The small scales of the estimated turbulent field are predicted more faithfully with latent-space DA, greatly reducing erroneous small-scale velocities typically introduced by state-space DA. Furthermore, latent-space DA is demonstrated to be robust to noisy measurements at the range of Reynolds numbers considered. These findings demonstrate that the observability of the system from available data can be greatly improved when turbulent measurements are assimilated in the right space, or coordinates.