  We develop a combinatorial framework to study certain polyhedral maps which
are higher-dimensional analogues of tropical covers between metric graphs.
Under a mild combinatorial assumption, we show that a map satisfies the
so-called balancing condition if and only if it is an indexed branched cover,
i.e.~locally over connected sets the count with multiplicity of points in every
fibre is a constant, which in particular gives a well-defined global degree
when the target is connected. Given a balanced map $(\Sigma, m_\Sigma) \to
\Delta$, we lift several connectivity properties of $\Delta$ to~$\Sigma$. Using
these lifting results we determine whether a multiplicity $m_{\mathcal U}$ that
is defined only on the interiors of maximal cells of $\Sigma$ can be extended
to all $\Sigma$ in a balanced manner. This relies on a strong connectivity
assumption; we give a counterexample when this is missing.
