This is a problem of Roger A.Horn's textbook: Matrix analysis, which is in this section 3.1. The problem P3. I will state this problem for you.
Suppose that $\mathbf{A} \in M_n$ has some non-real entries, but only real eigenvalue. Show that $\mathbf{A}$ is similar to a real matrix. Can the similarity matrix ever be chosen to be real?
What confused me is that I don't know how to prove this problem. If the claim doesn't hold, can you show me an couterexample.
The matrix $A$ is similar to its normal Jordan form $J$, which is a real matrix (the entries of the main diagonal of $J$ are the eigenvalues of $A$ and all other entries are $0$'s and $1$'s.
And, if $A=PJP^{-1}$, then, not, $P$ cannot be real. Because, if it was, then, since $J$ is real, $A$ would be real too.