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We propose a data-driven algorithm for reconstructing the irregular, chaotic flow dynamics around two side-by-side square cylinders from sparse, time-resolved, velocity measurements in the wake. We use Proper Orthogonal Decomposition (POD) to reduce the dimensionality of the problem and then explore two different reconstruction approaches: in the first approach, we use the subspace system identification algorithm n4sid to extract a linear dynamical model directly from the data (including the modelling and measurement error covariance matrices) and then employ Kalman filter theory to synthesize a linearly optimal estimator. In the second approach, the estimator matrices are directly identified using n4sid. A systematic study reveals that the first strategy outperforms the second in terms of reconstruction accuracy, robustness and computational efficiency. We also consider the problem of sensor placement. A greedy approach based on the QR pivoting algorithm is compared against sensors placed at the POD mode peaks; we show that the former approach is more accurate in recovering the flow characteristics away from the cylinders. We demonstrate that a linear dynamic model with a sufficiently large number of states and relatively few measurements, can recover accurately complex flow features, such as the interaction of the irregular flapping motion of the jet emanating from the gap with the vortices shed from the cylinders as well as the convoluted patterns downstream arising from the amalgamation of the individual wakes. The proposed methodology is entirely data-driven, does not have tunable parameters, and the resulting matrices are unique (to within a linear coordinate transformation of the state vector). The method can be applied directly to either experimental or computational data.