  Using large-scale density-matrix renormalzation group calculations and
minimally augmented spin-wave theory, we demonstrate that the phase diagram of
the quantum $S\!=\!\frac12$ $J_1$--$J_3$ ferro-antiferromagnetic model on the
honeycomb lattice differs dramatically from the classical one. It hosts the
double-zigzag and Ising-z phases as unexpected intermediaries between
ferromagnetic and zigzag states that are also extended beyond their classical
regions of stability. In broad agreement with quantum order-by-disorder
arguments, these collinear phases replace the classical spiral state.
