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We consider interacting (2+1)-dimensional Dirac fermions with competing symmetry-breaking electronic instabilities, as described by relativistic quantum field theories of the Gross-Neveu-Yukawa flavor with anticommuting mass terms. We demonstrate, using a combination of nonperturbative field-theoretical analysis and an adapted quantum Monte Carlo approach, that such systems exhibit a strong-coupling quantum multicritical fixed point with an emerging enhanced symmetry. Moreover, an extended phase coexistence regime expands out from this high-symmetry point. Our results disagree with recent results on the presence of a deconfined quantum criticality in (2+1)-dimensional Dirac fermions for the particular case of O(3) Néel and Z_{2} Kekulé symmetry-breaking instabilities on the graphene lattice. The robustness of these phenomena with respect to the microscopic symmetries furthermore demonstrates their relevance for a wide range of Dirac materials of current interest, from both theory and ongoing experiments.