  We relate full exceptional sequences in Fukaya categories of surfaces or
equivalently in derived categories of graded gentle algebras to branched
coverings over the disk, building on a previous classification result of the
first and third author. This allows us to apply tools from the theory of
branched coverings such as Birman--Hilden theory and Hurwitz systems to study
the natural braid group action on exceptional sequences. As an application,
counterexamples are given to a conjecture of Bondal--Polishchuk on the
transitivity of the braid group action on full exceptional sequences in a
triangulated category.
