  In evolutionary biology, it is common to study how various entities evolve
together, for example, how parasites coevolve with their host, or genes with
their species. Coevolution is commonly modelled by considering certain maps or
reconciliations from one evolutionary tree $P$ to another $H$, all of which
induce the same map $\phi$ between the leaf-sets of $P$ and $H$ (corresponding
to present-day associations). Recently, there has been much interest in
studying spaces of reconciliations, which arise by defining some metric $d$ on
the set $Rec(P,H,\phi)$ of all possible reconciliations between $P$ and $H$.
  In this paper, we study the following question: How do we compute a geometric
median for a given subset $\Psi$ of $Rec(P,H,\phi)$ relative to $d$, i.e. an
element $\psi_{med} \in Rec(P,H,\phi)$ such that $$ \sum_{\psi' \in \Psi}
d(\psi_{med},\psi') \le \sum_{\psi' \in \Psi} d(\psi,\psi') $$ holds for all
$\psi \in Rec(P,H,\phi)$? For a model where so-called host-switches or
transfers are not allowed, and for a commonly used metric $d$ called the
edit-distance, we show that although the cardinality of $Rec(P,H,\phi)$ can be
super-exponential, it is still possible to compute a geometric median for a set
$\Psi$ in $Rec(P,H,\phi)$ in polynomial time. We expect that this result could
be useful for computing a summary or consensus for a set of reconciliations
(e.g. for a set of suboptimal reconciliations).
