In our previous work, we proposed and experimentally demonstrated a physics-based simulator for combinatorial optimization problems (SSBM), which exploits a distinctive spontaneous symmetry breaking (SSB) phenomenon identified in a dissipative system exhibiting robust causality described by an iterative equation. We further conducted numerical simulations in which initial fluctuations of distinct patterns were applied to large-scale benchmark problems, and reported that the SSBM exhibits a unique behavior absent in conventional Ising machines: under specific conditions, pseudo-spin patterns progressively converge during the solution-search process and ultimately coalesce into a single pattern, with the cut value of the converged solution reaching 99.7% of the best-known cut value. In the present paper, we elucidate the conditions under which the SSBM can be configured to emulate the behavior of a continuous-variable Ising machine. We further demonstrate that by additionally incorporating an effect analogous to physical annealing into these conditions, the SSBM becomes capable of identifying the best-known cut value for the K₂₀₀₀ benchmark instance.