(2.2.3) For every $dlt$-model $\mathscr{X}$ of $X$, we can define the dual complex $\mathcal{D}((\mathscr{X}_{s})_{\mathrm{red}})$ for the $dlt$-pair $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ by gluing cells corresponding to irreducible components of intersections of irreducible components of $\mathscr{X}_{s}$, as in Definition 8 in [dFKX12]. When $k$ is not algebraically closed, we note that we only glue cells corresponding to irreducible components (instead of geometrically irreducible components). In other words, $\mathcal{D}((\mathscr{X}_{s})_{\mathrm{red}})$ is the quotient of the $\mathrm{Gal}(\bar{k}/k)$-equivariant dual complex constructed in [dFKX12, §31].