Are all normal maps antinormal?

Determine whether there exists a pointed category in which some normal map does not admit an antinormal decomposition; if so, develop sufficient conditions characterizing pointed categories in which every normal map admits an antinormal decomposition.

Background

A normal map factors as a normal epi followed by a normal mono; an antinormal decomposition reverses the order. It is unclear whether every normal map admits an antinormal factorization in general pointed categories.