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 $\mathrm{GL}_{n}$.