I'll recall the definition of the Gelfand-Zeitlin (extended eigenvalue) map $\gamma$ for Hermitian and upper-triangular $n$ by $n$ matrices. It defines a completely integrable Hamiltonian system with respect to the standard Poisson structures on these spaces. Using the coordinate system defined by a certain planar network $N$, we define a "tropical" analogue $\gamma_{trop}$ of the Gelfand-Zeiltin map. This is a piece-wise linear transformation of $\mathbb{R}^{n(n+1)/2}$ with interesting combinatorial properties described in terms of multiple paths on $N$. We establish a relation between fibers of $\gamma$ and $\gamma_{trop}$ and show that $\gamma_{trop}$ defines an open dense Darboux chart. (Joint work with I. Davydenkova, M. Podkopaeva and A. Szenes)