  In this article we introduce the $m$-cover poset of an arbitrary bounded
poset $\mathcal{P}$, which is a certain subposet of the $m$-fold direct product
of $\mathcal{P}$ with itself. Its ground set consists of multichains of
$\mathcal{P}$ that contain at most three different elements, one of which has
to be the least element of $\mathcal{P}$, and the other two elements have to
form a cover relation in $\mathcal{P}$. We study the $m$-cover poset from a
structural and topological point of view. In particular, we characterize the
posets whose $m$-cover poset is a lattice for all $m>0$, and we characterize
the special cases, where these lattices are EL-shellable, left-modular, or
trim. Subsequently, we investigate the $m$-cover poset of the Tamari lattice
$\mathcal{T}_{n}$, and we show that the smallest lattice that contains the
$m$-cover poset of $\mathcal{T}_{n}$ is isomorphic to the $m$-Tamari lattice
$\mathcal{T}_{n}^{(m)}$ introduced by Bergeron and Pr\'eville-Ratelle. We
conclude this article with a conjectural desription of an explicit realization
of $\mathcal{T}_{n}^{(m)}$ in terms of $m$-tuples of Dyck paths.
