 Real similarity from complex similarity

If real matrices satisfy $C=SD{S}^{-1}$ for an invertible complex $S=X+iY$, then $CS=SD$, so both real matrices $X,Y$ intertwine $C$ and $D$. Since $\det (X+tY)$ is a nonzero real polynomial, some real $t$ makes $X+tY$ invertible and supplies a real similarity.