Here's an example of $\Bbb{R}[[q]]$-total positivity arising from specializing an instance of Schur-positivity related to the Grassmannian:

Given an ordered $k$-subset $I = \{i_1 < \dots < i_k \}$
taken from $\{1, \dots, n\}$ let $s_I$ denote the Schur function associated to the partition $\big(i_k - k, \, \dots, \, i_1 - 1 \big)$ and let $[I]$ denote the corresponding Plücker coordinate on Grassmannian $\mathrm{Gr}(k,n)$.
The mapping $[I] \mapsto s_I$ extends to a ring homomorphsim from the homogeneous coordinate ring $\Bbb{C}\Big[\widehat{\mathrm{Gr}}(k,n) \Big]$ of the (affine cone over of the) Grassmannian $\mathrm{Gr}(k,n)$ to the ring of symmetric functions. Now take the principal specialization $x_i = q^{i-1}$ for $i\geq 1$ of the Schur functions. In this way we obtain a point $\mathrm{P}(q)$ in $\widehat{\mathrm{Gr}}(k,n)$ with the property that

\begin{equation}
\begin{array}{ll}
\displaystyle [I](\mathrm{P}(q)) 
&\displaystyle = \, s_I \big(1,q,q^2, \dots \big) \\ \\
&\displaystyle = \,
q^{\eta(\lambda)} \, \prod_{\stackrel{\scriptstyle\mathrm{boxes}}{b \in \lambda}} \, \Big( 1 - q^{ \, \mathrm{hook}(b)} \Big)^{-1}
\end{array}
\end{equation}

which is clearly $\Bbb{R}[[q]]$-totally positive upon expansion.

regards, ines.