  The ordering of the three-dimensional Heisenberg spin glass with the weak
random anisotropy in magnetic fields is studied by extensive equilibrium Monte
Carlo simulations. Both the spin and the chirality are monitored. We find
strong numerical evidence that a replica symmetry breaking transition occurs in
the chiral sector, which accompanies the simultaneous spin-glass order. Despite
the similarity in the global symmetry, the ordering behavior of the weakly
anisotropic Heisenberg spin glass differs significantly from that of the
strongly anisotropic Ising spin glass. The obtained phase diagram in the
temperature - magnetic field plane is similar to the experimental phase
diagram. Our results highlight the importance of the chirality in the
spin-glass ordering of the Heisenberg-like spin glass, and support the
spin-chirality decoupling-recoupling scenario of spin-glass transitions.
