  A quasi-exponential is an entire function of the form $e^{cu}p(u)$, where
$p(u)$ is a polynomial and $c \in \mathbb{C}$. Let $V = \langle e^{h_1u}p_1(u),
\dots, e^{h_Nu}p_N(u) \rangle$ be a vector space with a basis of
quasi-exponentials. We show that if $h_1, \dots, h_N$ are nonnegative and all
of the complex zeros of the Wronskian $\operatorname{Wr}(V)$ are real, then $V$
is totally nonnegative in the sense that all of its Grassmann-Pl\"{u}cker
coordinates defined by the Taylor expansion about $u=t$ are nonnegative, for
any real $t$ greater than all of the zeros of $\operatorname{Wr}(V)$. Our proof
proceeds by showing that the higher Gaudin Hamiltonians $T_\lambda^G(t)$
introduced in [ALTZ14] are universal Pl\"ucker coordinates about $u=t$ for the
Wronski map on spaces of quasi-exponentials. The result that $V$ is totally
nonnegative follows from the fact that $T_\lambda^G(t)$ is positive
semidefinite, which we establish using partial traces. We also show that if
$h_1 = \cdots = h_N = 0$ then $T_\lambda^G(t)$ equals $\beta^\lambda(t)$, which
is the universal Pl\"ucker coordinate for the Wronski map on spaces of
polynomials introduced in [KP23].
