  We show that branched coverings of surfaces of large enough genus arise as
characteristic maps of braided surfaces that is, lift to embeddings in the
product of the surface with $\mathbb R^2$. This result is nontrivial already
for unramified coverings, in which case the lifting problem is well-known to
reduce to the purely algebraic problem of factoring the monodromy map to the
symmetric group $S_n$ through the braid group $B_n$. In our approach, this
factorization is often achieved as a consequence of a stronger property: a
factorization through a free group. In the reverse direction we show that any
non-abelian surface group has infinitely many finite simple non-abelian groups
quotients with characteristic kernels which do not contain any simple loop and
hence the quotient maps do not factor through free groups. By a pullback
construction, finite dimensional Hermitian representations of braid groups
provide invariants for the braided surfaces. We show that the strong
equivalence classes of braided surfaces are separated by such invariants if and
only if they are profinitely separated.
