  We show that the Ginzburg-Weinstein diffeomorphism $\mathfrak{u}(n)^* \to
U(n)^*$ of Alekseev-Meinrenken admits a scaling tropical limit on an open dense
subset of $\mathfrak{u}(n)^*$. The target of the limit map is a product
$\mathcal{C} \times T$, where $\mathcal{C}$ is the interior of a cone, $T$ is a
torus, and $\mathcal{C} \times T$ carries an integrable system with natural
action-angle coordinates. The pull-back of these coordinates to
$\mathfrak{u}(n)^*$ recovers the Gelfand-Zeitlin integrable system of
Guillemin-Sternberg. As a by-product of our proof, we show that the Lagrangian
tori of the Flaschka-Ratiu integrable system on the set of upper triangular
matrices meet the set of totally positive matrices for sufficiently large
action coordinates.
