**Preliminaries**

A complex matrix $A$ is *normal* when $A$ and $A^*$ commute. A real matrix $A$ is *normal* when $A$ and $A^t$ commute.

Two complex matrices $A$ and $B$ are said to be *unitary similar* if there exists a unitary matrix $U$ such that $A\cdot U=U\cdot B$. Two real matrices $A$ and $B$ are *orthogonal similar* if there exists a (real) orthogonal matrix $O$ such that $A\cdot O=O\cdot B$.

When $A$ and $B$ are complex *normal* matrices then $A$ and $B$ are unitary similar if and only if $A$ and $B$ have the same characteristic polynomial (see e.g., this post).

Let $A$ and $B$ be real normal matrices. If $A$ and $B$ are orthogonal similar then $A$ and $B$ have the same characteristic polynomial. The converse does not hold, however, since $A$ and $B$ may have complex eigenvalues and unitary rather than orthogonal matrices are needed.

**Question**
I would like to have an *graph-based* example showing that having the same characteristic polynomial does not suffice for orthogonal similarity.

More precisely, call a *directed* graph $G=(V,E)$ *normal* if its adjacency matrix $A_G$ is normal.

Normal directed graphs are necessarily *balanced*, i.e., the in-degree of each vertex is equal to its out-degree (see e.g., this post).

So, what are examples of two normal (and thus balanced) graphs $G$ and $H$ (consisting of the same number of vertices) whose adjacency matrices have the same characteristic polynomial yet are not related by means of an orthogonal similarity?