Suppose $A,E$ are Hermitian $(n \times n)$-matrices and $E$ is of low rank. There are well known results bounding the difference in spectra of $A$ and $A+E$. For example the Eigenvalue Interlacing Theorem states \begin{align*} & \lambda_{j-\operatorname{rank}(E)}(A) \geq \lambda_{j}(A+E) \geq \lambda_{j+\operatorname{rank}(E)}(A) \end{align*} for $\lambda_1(M) \geq ... \geq \lambda_n(M)$, whenever the eigenvalues with the corresponding indices exist.

Question: Can we do something similar when $A$ is not Hermitian, but still has real eigenvalues? Let $A,E$ be (not necessarily Hermitian) $(n \times n)$-matrices such that the eigenvalues of $A$ and $A+E$ are all real. Is there a link or bound on the difference between the eigenvalues of $A$ and those of $A+E$ for low rank $E$?

[If it helps, we may assume the eigenvalues of $A$ to be positive, instead of just real.]

The standard counter example for low-rank perturbation having large effects on the eigenvalues of non-Hermitian matrices is $\tilde{A} = (\delta_{k+1,l})_{k,l \leq n}$ and $\tilde{E}_{k,l}=\begin{cases} 1 & \text{ for } k=1, \, l=n \\ 0 & \text{ else} \end{cases}$, since $\tilde{A}$ has only the eigenvalue $0$ with multiplicity $n$ and $(\tilde{A}+\tilde{E})$ has eigenvalues $(e^{i2\pi \frac{j}{n}})_{j \leq n}$. A similar example, where all eigenvalues are in $\mathbb{R}$ would answer my question in the negative.

Any help is much appreciated!