 Solution

Take two directed triangles meeting only at a central vertex $a$. Make $a\to b\to c\to a$ entirely black and $a\to d\to e\to a$ entirely white. All five vertices lie in one strongly connected component, since every vertex can reach $a$ and can be reached from $a$. It contains internal edges of both colours.

A closed walk can traverse the black triangle and then the white triangle. But a simple cycle using edges from both triangles would have to pass through $a$ to change triangles and pass through $a$ again to return. It would repeat $a$ other than the single start/end identification, contradicting simplicity. Removing $a$ separates the two triangles, so no other transition between them is possible. ThereforeThis example avoids self-loops and works even when a directed simple cycle is required to have at least three vertices.