A Tannakian description of the local Kaletha gerbe

We construct, for a $p$-adic field $F$, an explicit semisimple Tannakian category ${\text{RigIsoc}}_F$ whose category of fiber functors recovers Kaletha’s Galois gerbe ${\mathcal{E}}_{\text{Kal}}$. We then classify and write down the simple objects in ${\text{RigIsoc}}_F$, all of which come from elliptic twisted Levi subgroups of ${\mathrm{G}\mathrm{L}}_n$.