A Nordhaus–Gaddum-type result is a lower or an upper bound on the sum or the product of a parameter of a graph and its complement. In this paper we continue the study of Nordhaus–Gaddum bounds for the total Roman domination number $$\gamma _{tR}$$ . Let G be a graph on n vertices and let $$\overline{G}$$ denote the complement of G, and let $$\delta ^*(G)$$ denote the minimum degree among all vertices in G and $$\overline{G}$$ . For $$\delta ^*(G)\ge 1$$ , we show that (i) if G and $$\overline{G}$$ are connected, then $$(\gamma _{tR}(G)-4)(\gamma _{tR}(\overline{G})-4)\le 4\delta ^*(G)-4$$ , (ii) if $$\gamma _{tR}(G), \gamma _{tR}(\overline{G})\ge 8$$ , then $$\gamma _{tR}(G)+\gamma _{tR}(\overline{G})\le 2\delta ^*(G)+5$$ and (iii) $$\gamma _{tR}(G)+\gamma _{tR}(\overline{G})\le n+5$$ and $$\gamma _{tR}(G)\gamma _{tR}(\overline{G})\le 6n-5$$ .