I think I found something! After long and hard googling, I stumbled upon this:

MR2311036 (2008e:15011) 15A18 (15A29 15A48)
Chen, Jianbiao; Xu, Zhaoliang The inverse eigenvalue problem for real eventually positive matrices. Filomat 21 (2007), no. 1, 1–16.

http://www.doiserbia.nb.rs/img/doi/0354-5180/2007/0354-51800701001C.pdf

The result that interests me is a structure theorem for eventually positive matrices (Theorem 3.2):

A real $n \times n$ matrix $A$ is eventually positive iff there exist positive vectors $\alpha,\beta$ (of length $n$) and a $n \times n$ matrix $Y$ so that:

$A=\frac{1}{(\beta^{T}\alpha)^{2})}\alpha\beta^{T}+\frac{1}{\beta^{T}\alpha}(I-\frac{1}{\beta^{T}\alpha}\alpha\beta^{T})Y(I-\frac{1}{\beta^{T}\alpha}\alpha\beta^{T})$ and the spectral radius of $(I-\frac{1}{\beta^{T}\alpha}\alpha\beta^{T})Y(I-\frac{1}{\beta^{T}\alpha}\alpha\beta^{T})$ is $<1$.