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-RP3 model), encompasses Ising (chiral) degrees of freedom, spin-wave excitations, and Z2 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 Z2 vortices proliferate near the chiral domain walls, explaining the strong renormalization of the transition temperature. Chiral domain walls can themselves carry an unlocalized Z2 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-RP3 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-RP3 model.