Question: I am wondering whether there exists a sufficient and necessary condition that has to be obeyed by a non-negative $n\times n$ matrix $A=\{a_{ij}\}\in \mathbb{R}^{n \times n}$, which guarantees that $A$ has only real eigenvalues (I don't mind whether they are positive, negative, zero; or any combination of these) but no complex ones.
Of course, symmetric matrices have real spectra -- but this is not a necessary condition, since certain non-symmetric matrices have real spectra too. For example, the non-negative matrices $A(\lambda) = \begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix}$ with $\lambda \geq 0$ are not symmetric, yet they have a purely real spectrum (namely the eigenvalue $\lambda$). On the other hand, there are known examples showing that arbitrary complex number $a+ib$ is an eigenvalue of some non-negative square matrix, see e.g. the beautiful example here.
Example: The matrices I am trying to understand are too large too display. However, to give one concrete example, consider the family of matrices $$ A(t,\gamma) = \left( \begin{array}{cccccccccccc} 0 & \gamma +t & \gamma +t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ t-\gamma & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & t-\gamma & 0 & 0 \\ t-\gamma & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & t-\gamma & 0 \\ 0 & 0 & 0 & 0 & 0 & \gamma +t & 0 & \gamma +t & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \gamma +t & 0 & 0 & \gamma +t & 0 & 0 & 0 \\ 0 & 0 & 0 & t-\gamma & t-\gamma & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \gamma +t & \gamma +t & 0 & 0 & 0 \\ 0 & 0 & 0 & t-\gamma & 0 & 0 & t-\gamma & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & t-\gamma & 0 & t-\gamma & 0 & 0 & 0 & 0 & 0 \\ 0 & \gamma +t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \gamma +t \\ 0 & 0 & \gamma +t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \gamma +t \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & t-\gamma & t-\gamma & 0 \\ \end{array} \right) $$ where $t$ and $\gamma$ are real coefficients. The matrix $A(t,\gamma)$ does not seem to have any special neat properties (for example, it is not symmetric nor stochastic). Nevertheless, one can easily verify that whenever the matrix is non-negative (that is, when $t>\gamma>-t$), it has real spectrum. (It also happens to have a purely imaginary spectrum whenever these inequalities do not hold, and a purely zero spectrum when $\gamma=\pm t$.)
Does $A(t,\gamma)$ have some special property that I overlook, which guarantees the reality of the spectrum (and which I could use to generate other similar matrices with real spectrum)? Or is there some efficient algorithmic check of the reality of the spectrum that could be run on such a class of non-negative matrices without calculating the spectrum explicitly?
