  Phylogenetic networks are rooted, labelled directed acyclic graphs which are
commonly used to represent reticulate evolution. There is a close relationship
between phylogenetic networks and multi-labelled trees (MUL-trees). Indeed, any
phylogenetic network $N$ can be 'unfolded' to obtain a MUL-tree $U(N)$ and,
conversely, a MUL-tree $T$ can in certain circumstances be 'folded' to obtain a
phylogenetic network $F(T)$ that exhibits $T$. In this paper, we study
properties of the operations $U$ and $F$ in more detail. In particular, we
introduce the class of stable networks, phylogenetic networks $N$ for which
$F(U(N))$ is isomorphic to $N$, characterise such networks, and show that that
they are related to the well-known class of tree-sibling networks. We also
explore how the concept of displaying a tree in a network $N$ can be related to
displaying the tree in the MUL-tree $U(N)$. To do this, we develop a
phylogenetic analogue of graph fibrations. This allows us to view $U(N)$ as the
analogue of the universal cover of a digraph, and to establish a close
connection between displaying trees in $U(N)$ and reconciling phylogenetic
trees with networks.
