In the first part of this article we present a realization of the m-Tamari lattice T(m)n in terms of m-tuples of Dyck paths of height n, equipped with componentwise rotation order. For that, we define the m-cover poset P⟨m⟩ of an arbitrary bounded poset P, and show that the smallest lattice completion of the m-cover poset of the Tamari lattice Tn is isomorphic to the m-Tamari lattice T(m)n. A crucial tool for the proof of this isomorphism is a decomposition of m-Dyck paths into m-tuples of classical Dyck paths, which we call the strip-decomposition. Subsequently, we characterize the cases where the m-cover poset of an arbitrary poset is a lattice. Finally, we show that the m-cover poset of the Cambrian lattice of the dihedral group is a trim lattice with cardinality equal to the generalized Fuss-Catalan number of the dihedral group.