  We study the two-dimensional fully frustrated XY (FFXY) model and two related
models, a discretization of the Landau-Ginzburg-Wilson Hamiltonian for the
critical modes of the FFXY model and a coupled Ising-XY model, by means of
Monte Carlo simulations on square lattices L x L, L=O(10^3). We show that their
phase diagram is characterized by two very close chiral and spin transitions,
at T_ch > T_sp respectively, of the Ising and Kosterlitz-Thouless type. At T_ch
the Ising regime sets in only after a preasymptotic regime, which appears
universal to some extent. The approach is nonmonotonic for most observables,
with a wide region controlled by an effective exponent nu_eff=0.8.
