This question is a follow-up and an improvement (I hope), to Is the dual graph simple?
According to the book Topological Graph Theory by Gross and Tucker, given a cellular embedding of a graph on a surface (by 'surface' I mean here a sphere with $n\geq 0$ handles), one can define a dual multigraph by treating the faces of the original graph embedding as vertices and adding an edge between two vertices for every edge the corresponding faces have in common in the original graph.
Let $G$ be cellularly embedded in some surface of genus $n$ and let $G'$ be the dual of this embedding.
What conditions on the embedding of $G$ (necessary and/or sufficient) ensure that $G'$ is a simple graph (at most one edge between two vertices and no loops)?