  Phylogenomics commonly aims to construct evolutionary trees from genomic
sequence information. One way to approach this problem is to first estimate
event-labeled gene trees (i.e., rooted trees whose non-leaf vertices are
labeled by speciation or gene duplication events), and to then look for a
species tree which can be reconciled with this tree through a
\emph{reconciliation map} between the trees. In practice, however, it can
happen that there is no such map from a given event-labeled tree to \emph{any}
species tree. An important situation where this might arise is where the
species evolution is better represented by a \emph{network} instead of a tree.
In this paper, we therefore consider the problem of reconciling event-labeled
trees with species networks. In particular, we prove that any event-labeled
gene tree can be reconciled with some network and that, under certain mild
assumptions on the gene tree, the network can even be assumed to be multi-arc
free. To prove this result, we show that we can always reconcile the gene tree
with some multi-labeled (MUL-)tree, which can then be "folded up" to produce
the desired reconciliation and network. In addition, we study the interplay
between reconciliation maps from event-labeled gene trees to MUL-trees and
networks. Our results could be useful for understanding how genomes have
evolved after undergoing complex evolutionary events such as polyploidy.
