Simple right-alternative unital superalgebras over an algebra of matrices of order $2$

We classify simple right-alternative unital superalgebras over a field of characteristic not $2$, whose even part coincides with an algebra of matrices of order $2$. It is proved that such a superalgebra either is a Wall double $W_{2|2}(\omega)$, or is a Shestakov super algebra $S_{4|2}(\sigma)$ (characteristic $3$), or is isomorphic to an asymmetric double, an $8$-dimensional superalgebra depending on four parameters. In the case of an algebraically closed base field, every such superalgebra is isomorphic to an associative Wall double $\mathrm{M}_2[\sqrt{1}]$, an alternative $6$-dimensional Shestakov superalgebra $B_{4|2}$ (characteristic $3$), or an $8$-dimensional Silva–Murakami–Shestakov superalgebra.