  We drive a d-dimensional Heisenberg magnet using a spatially anisotropic
current of mobile particles or heat. The continuum Langevin equation is
analyzed using a dynamical renormalization group, stability analysis and
numerical simulations. We discover a rich steady-state phase diagram, including
a critical point in a new nonequilibrium universality class, and a
spatiotemporally chaotic phase. The latter may be `controlled' in a robust
manner to target spatially periodic steady states with helical order. We
discuss several physical realizations of this model and make definite
predictions which could be tested in experimental or model lattice systems.
