  We describe a systematic way of the generalization, to models with non-linear
duality, of the space-time covariant and duality-invariant formulation of
duality-symmetric theories in which the covariance of the action is ensured by
the presence of a single auxiliary scalar field. It is shown that the
duality-symmetric action should be invariant under the two local symmetries
characteristic of this approach, which impose constraints on the form of the
action similar to those of Gaillard and Zumino and in the non-covariant
formalism. We show that the (twisted) self-duality condition obtained from this
action upon integrating its equations of motion can always be recast in a
manifestly covariant form which is independent of the auxiliary scalar and thus
corresponds to the conventional on-shell duality-symmetric covariant
description of the same model. Supersymmetrization of this construction is
briefly discussed.
