  We present a theoretical study of field-induced magnetic phases in the
honeycomb Kitaev-Heisenberg model, which is believed to describe the essential
physics of Mott insulators with strong spin-orbit coupling such as $A_2$IrO$_3$
and $\alpha$-RuCl$_3$. We obtain a finite temperature phase diagram based on
extensive Monte Carlo simulations and analytical calculations. We show that,
while Zeeman coupling favors a symmetric non-coplanar magnetic order, thermal
fluctuations enhances the stability of a collinear zigzag phase that breaks the
rotational symmetry of the lattice. Our large-scale simulations also uncover
intriguing commensurate-incommensurate transitions and multiple-$\mathbf Q$
incommensurate phases at high field. Experimental implications are also
discussed.
