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Despite being at the heart of the theory of the "Big Bang" and cosmic inflation, the quantum field theory prediction of false vacuum tunneling has not been tested. To address the exponential complexity of the problem, a table-top quantum simulator in the form of an engineered Bose-Einstein condensate (BEC) has been proposed to give dynamical solutions of the quantum field equations. In this paper, we give a numerical feasibility study of the BEC quantum simulator under realistic conditions and temperatures, with an approximate truncated Wigner (tW) phase-space method. We report the observation of false vacuum tunneling in these simulations, and the formation of multiple bubble 'universes' with distinct topological properties. The tunneling gives a transition of the relative phase of coupled Bose fields from a metastable to a stable 'vacuum'. We include finite temperature effects that would be found in a laboratory experiment and also analyze the cut-off dependence of modulational instabilities in Floquet space. Our numerical phase-space model does not use thin-wall approximations, which are inapplicable to cosmologically interesting models. It is expected to give the correct quantum treatment including superpositions and entanglement during dynamics. By analyzing a nonlocal observable called the topological phase entropy (TPE), our simulations provide information about phase structure in the true vacuum. We observe a cooperative effect in which the true vacua bubbles representing distinct universes each have one or the other of two distinct topologies. The TPE initially increases with time, reaching a peak as the multiple universes are formed, and then decreases with time to the phase-ordered vacuum state. This gives a model for the formation of universes with one of two distinct phases, which is a possible solution to the problem of particle-antiparticle asymmetry.