  We present an alternative formulation of duality-symmetric eleven-dimensional
supergravity with both three-form and six-form gauge fields. Instead of the
recently-proposed scalar auxiliary field, we use a simpler lagrangian with a
non-propagating auxiliary multiplier tensor field with eight-indices. We also
complete the superspace formulation in a duality-symmetric manner. An
alternative super M-5-brane action coupled to this eleven-dimensional
background is also presented. This formulation bypasses the usual obstruction
for an invariant lagrangian for a self-dual three-form field strength, by
allowing the self-duality only as a solution for field equations, but not as a
necessary condition.
