Data assimilation integrates observational data with predictive simulation models to produce coherent estimates of the full state of complex physical systems. However, simulation models inevitably neglect small-scale processes, are sensitive to perturbations, or oversimplify parameter correlations, leading to reconstructions that diverge from sensor measurements of real physics. This creates a critical simulation-to-reality (SIM2REAL) gap. We propose DA-SHRED (Data Assimilation with SHallow REcurrent Decoder), a machine learning framework that bridges the SIM2REAL gap by leveraging the latent space learned from reduced simulation models via the SHRED architecture. SHRED exploits three key concepts: separation of variables, Takens embedding theorem, and a decoding-only strategy to reconstruct full state-space dynamics from sparse sensor measurements. Our framework updates latent variables using real sensor data to accurately reconstruct full system states that cannot be directly observed. Furthermore, DA-SHRED incorporates a sparse identification of nonlinear dynamics (SINDy) regression model in the latent space to identify functionals corresponding to missing dynamics in simulation models. Two SINDy-based algorithms are presented for extracting parsimonious missing physics from libraries of candidate terms. The method is demonstrated on four challenging systems: 2D damped Kuramoto-Sivashinsky equations, 2D Kolmogorov flow, 2D Gray-Scott reaction-diffusion equations, and rotating detonation engines. Results show that DA-SHRED closes the SIM2REAL gap by approximately an order of magnitude in error within short evolution windows while successfully recovering missing physics terms. Theoretical analysis establishes that DA-SHRED preserves representational structure under suitable assumptions, with port-Hamiltonian systems providing rigorous examples of eigenspace preservation under perturbations. The algorithm is robust to noise, requires minimal training data, and enables efficient laptop-level computing. DA-SHRED demonstrates a powerful synergy between data assimilation, reduced-order modeling, and physics-informed model discovery for real-time, data-efficient state estimation in scientific and engineering applications.