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 St E of G(E) is then χ-distinguished for a given unique character χ of G(F), and that χ occurs with multiplicity 1 in the restriction of St E to G(F). In the first two papers of the series, Broussous and the author have proved the Prasad conjecture when G is F-split and E/F is unramified; this paper deals with the tamely ramified case, still with G F-split.