  In this paper, we study robust transshipment under consistent flow
constraints. We consider demand uncertainty represented by a finite set of
scenarios and characterize a subset of arcs as so-called fixed arcs. In each
scenario, we require an integral flow that satisfies the respective flow
balance constraints. In addition, on each fixed arc, we require equal flow for
all scenarios. The objective is to minimize the maximum cost occurring among
all scenarios. We show that the problem is strongly NP-hard on acyclic digraphs
by a reduction from the $(3,B2)$-SAT problem. Furthermore, we prove that the
problem is weakly NP-hard on series-parallel digraphs by a reduction from a
special case of the \textsc{Partition} problem. If in addition the number of
scenarios is constant, we observe the pseudo-polynomial-time solvability of the
problem. We provide polynomial-time algorithms for three special cases on
series-parallel digraphs. Finally, we present a polynomial-time algorithm for
pearl digraphs.
