Similarity of real matrices over $\mathbb{C}$
If $ A \underset{\mathbb{C}}{\sim} B$, there is a matrix $C \in GL_n(\mathbb{C})$ such that $A=C^{-1}BC$.
So $CA=BC$.
$C=P+iQ$ with $P,Q \in M_n(\mathbb{R})$.
If $A \in M_n(\mathbb{R})$ and $B \in M_n(\mathbb{R})$, we have $CA=BC \implies (P+iQ)A=B(P+iQ)\implies PA=BP$ and $QA=BQ$.
The polynomial $\det (P+XQ)$ is not null, because $\det(P+iQ)=\det C \neq 0$.
So, there is a value $\lambda \in \mathbb{R}$ such hat $\det(P+\lambda Q) \neq 0$.
Let $D=P+\lambda Q$, $DA=BD$ because $PA=BP$ and $QA=BQ$.
So, $A=D^{-1}BD$ and $A \underset{\mathbb{R}}{\sim} B$