  We investigate the ground-state properties of the highly degenerate
non-coplanar phase of the classical bilinear-biquadratic Heisenberg model on
the triangular lattice with Monte Carlo simulations. For that purpose, we
introduce an Ising pseudospin representation of the ground states, and we use a
simple Metropolis algorithm with local updates, as well as a powerful cluster
algorithm. At sizes that can be sampled with local updates, the presence of
long-range order is surprisingly combined with an algebraic decay of
correlations and the complete disordering of the chirality. It is only thanks
to the investigation of unusually large systems (containing $\sim 10^8$ spins)
with cluster updates that the true asymptotic regime can be reached and that
the system can be proven to consist of equivalent (i.e., equally ordered)
sublattices. These large-scale simulations also demonstrate that the scalar
chirality exhibits long-range order at zero temperature, implying that the
system has to undergo a finite-temperature phase transition. Finally, we show
that the average distance in the order parameter space, which has the structure
of an infinite Cayley tree, remains remarkably small between any pair of
points, even in the limit when the real space distance between them tends to
infinity.
