  We construct a manifestly diffeomorphic M5-brane action in dual formulation
coupled to an eleven-dimensional supergravity target space. The
covariantisation is carried out by using (generalised) PST technique with 5
auxiliary scalar fields, which are obtained by using a geometrical
consideration as a reduction of an auxiliary 4-form of
Maznytsia-Preitschopf-Sorokin. As is typical in PST-covariantised theory, our
construction possesses as usual the two local PST symmetries. By using one of
the local PST symmetries, the action can be reduced to the non-manifestly
covariant M5-brane action in dual formulation constructed earlier by the
authors. The discussion on double dimensional reduction to D4-brane, and on the
comparison of on-shell action then easily follows.
