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Prior work has demonstrated that reconfigurable logic can significantly benefit certain applications. However, reconfigurable architectures have traditionally suffered from high area overhead and limited application coverage. We present a dynamically managed multithreaded reconfigurable architecture consisting of multiple clusters of shared reconfigurable fabrics that greatly reduces the area overhead of reconfigurability while still offering the same power efficiency and performance benefits. Like other shared SMT and CMP resources, the dynamic partitioning of the reconfigurable resource among sharing threads, along with the co-scheduling of threads among different reconfigurable clusters, must be intelligently managed for the full benefits of the shared fabrics to be realized. We propose a number of sophisticated dynamic management approaches, including the application of machine learning, multithreaded phase-based management, and stability detection. Overall, we show that, with our dynamic management policies, multithreaded reconfigurable fabrics can achieve better energy×delay2, at far less area and power, than providing each core with a much larger private fabric. Moreover, our approach achieves dramatically higher performance and energy-efficiency for particular workloads compared to what can be ideally achieved by allocating the fabric area to additional cores.