  We describe the complex global structure of giant components in directed
multiplex networks which generalizes the well-known bow-tie structure, generic
for ordinary directed networks. By definition, a directed multiplex network
contains vertices of one type and directed edges of $m$ different types. In
directed multiplex networks, we distinguish a set of different giant components
based on the existence of directed paths of different types between their
vertices, such that for each type of edges, the paths run entirely through only
edges of that type. If, in particular, $m=2$, we define a strongly viable
component as a set of vertices, in which for each type of edges, each two
vertices are interconnected by at least two directed paths in both directions,
running through the edges of only this type. We show that in this case, a
directed multiplex network contains, in total, $9$ different giant components
including the strongly viable component. In general, the total number of giant
components is $3^m$. For uncorrelated directed multiplex networks, we obtain
exactly the size and the emergence point of the strongly viable component and
estimate the sizes of other giant components.
