We introduce a minimal model describing the physics of classical two-dimensional (2D) frustrated Heisenberg systems, where spins order in a nonplanar way at $T=0$. This model, consisting of coupled trihedra (or Ising-$\mathbb{R}{P}^{3}$ model), encompasses Ising (chiral) degrees of freedom, spin-wave excitations, and ${\mathbb{Z}}_{2}$ vortices. Extensive Monte Carlo simulations show that the $T=0$ chiral order disappears at finite temperature in a continuous phase transition in the 2D Ising universality class, despite misleading intermediate-size effects observed at the transition. The analysis of configurations reveals that short-range spin fluctuations and ${\mathbb{Z}}_{2}$ vortices proliferate near the chiral domain walls, explaining the strong renormalization of the transition temperature. Chiral domain walls can themselves carry an unlocalized ${\mathbb{Z}}_{2}$ topological charge, and vortices are then preferentially paired with charged walls. Further, we conjecture that the anomalous size effects suggest the proximity of the present model to a tricritical point. A body of results is presented, which all support this claim: (i) first-order transitions obtained by Monte Carlo simulations on several related models, (ii) approximate mapping between the Ising-$\mathbb{R}{P}^{3}$ model and a dilute Ising model (exhibiting a tricritical point), and, finally, (iii) mean-field results obtained for Ising-multispin Hamiltonians, derived from the high-temperature expansion for the vector spins of the Ising-$\mathbb{R}{P}^{3}$ model.
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