In this paper, we propose a fully discrete finite element based discretization for the numerical approximation of the stochastic Allen‐Cahn‐Navier‐Stokes system on a bounded polygonal domain of
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. We prove that the proposed numerical scheme is unconditionally solvable, has finite energies and constructs weak martingale solutions of the stochastic Allen‐Cahn‐Navier‐Stokes...