Distinction of the Steinberg representation III: the tamely ramified case

Let $F$ be a nonarchimedean local field, let $E$ be a Galois quadratic extension of $F$ and let $G$ be a quasisplit group defined over $F$; a conjecture by Dipendra Prasad states that the Steinberg representation of $G(E)$ is then $\chi$-distinguished for a given unique character $\chi$ of $G(F)$. In the first two papers of the series, Broussous and the author have proved that result when $G$ is $F$-split and $E\mathrm{/}F$ is unramified; this paper deals with the tamely ramified case, still with $G$ $F$-split.