= Solution
Take
$$
\boxed{E=\begin{pmatrix}1&1\\0&2\end{pmatrix}.}
$$
The real <eigenvalues> are one and two, with eigenlines spanned by $(1,0)^T$ and $(1,1)^T$ respectively. These independent vectors form a real <eigenbasis>, so $E$ is a <diagonalizable matrix> over $\mathbb R$. Their <inner product> is one, and any two nonzero vectors on those eigenlines have nonzero <inner product>. Rescaling can normalize their lengths but cannot make them orthogonal. Hence no orthonormal <eigenbasis> exists.
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