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-Plücker 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 $T_λ^G(t)$ introduced in [ALTZ14] are universal Plücker 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 $T_λ^G(t)$ is positive semidefinite, which we establish using partial traces. We also show that if $h_1 = \cdots = h_N = 0$ then $T_λ^G(t)$ equals $β^λ(t)$, which is the universal Plücker coordinate for the Wronski map on spaces of polynomials introduced in [KP23].