Let $A_1, A_2,..., A_m$ be a collection of real $n \times n$ matrices.

What are some conditions (necessary or sufficient) on $A_1,...,A_m$ for any product $A_{i_1} A_{i_2} \cdots A_{i_k}$ to have non-negative trace?

One sufficient condition is if there is some matrix $P$ so that $PA_iP^{-1}$ has positive entries for each $i$.

However, this condition is not necessary: Let $A_1 = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ and $A_2 = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}.$ Then products of $A_1$ and $A_2$ are of the form $\begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix}$ for $a \in \mathbb{Z}$, and thus have trace 2. But there is no matrix $P$ so that $P A_1 P^{-1}$ and $PA_2 P^{-1}$ have positive entries.

This question is inspired by the paper Proof of a conjecture on immanants of the Jacobi-Trudi matrix, where Greene shows that irreducible characters evaluated on certain elements of $\mathbb{C}[\mathfrak{S}_n]$ are non-negative by showing that the corresponding elements of the irreducible representation can be written as matrices with positive entries.