  We consider a totally nonsymplectic Anosov action of Z^k which is either
uniformly quasiconformal or pinched on each coarse Lyapunov distribution. We
show that such an action on a torus is C^\infty--conjugate to an action by
affine automorphisms. We also obtain similar global rigidity results for
actions on an arbitrary compact manifold assuming that the coarse Lyapunov
foliations are jointly integrable.
