If $X_{i,j}$ is a $2 \times 4$ matrix and $Y_{j,k}$ is a $4 \times 2$ matrix, then there is a $P^1$ family of ideals defined by the four equations $XY = 0$ and the six equations $s \cdot minors(X) + t \cdot minors^{\vee}(Y) = 0$ where $[s:t]$ are the projective coordinates of $P^1$ and $minors^{\vee}$ is the complementary set of six minors. Explicitly,
\begin{align*} x_{ 0,0 }y_{ 0,0 }+x_{ 0,1 }y_{ 0,1 }+x_{ 0,2 }y_{ 0,2 }+x_{ 0,3 }y_{ 0,3 } &= 0 \\ x_{ 0,0 }y_{ 1,0 }+x_{ 0,1 }y_{ 1,1 }+x_{ 0,2 }y_{ 1,2 }+x_{ 0,3 }y_{ 1,3 } &= 0 \\ x_{ 1,0 }y_{ 0,0 }+x_{ 1,1 }y_{ 0,1 }+x_{ 1,2 }y_{ 0,2 }+x_{ 1,3 }y_{ 0,3 } &= 0 \\ x_{ 1,0 }y_{ 1,0 }+x_{ 1,1 }y_{ 1,1 }+x_{ 1,2 }y_{ 1,2 }+x_{ 1,3 }y_{ 1,3 } &= 0 \\ s(x_ { 1,0 }x_{ 0,1 }-x_{ 0,0 }x_{ 1,1 }) + t (-y_{ 1,2 }y_{ 0,3 }+y_{ 0,2 }y_{ 1,3 }) &= 0 \\ s( -x_{ 1,1 }x_{ 0,3 }+x_{ 0,1 }x_{ 1,3 }) + t(-y_{ 1,0 }y_{ 0,2 }+y_{ 0,0 }y_{ 1,2 }) &= 0\\ s( -x_{ 1,0 }x_{ 0,3 }+x_{ 0,0 }x_{ 1,3 })+t(y_{ 1,1 }y_{ 0,2 }-y_{ 0,1 }y_{ 1,2 }) & = 0 \\ s(-x_{ 1,1 }x_{ 0,2 }+x_{ 0,1 }x_{ 1,2 })+t(y_{ 1,0 }y_{ 0,3 }-y_{ 0,0 }y_{ 1,3 }) & = 0\\ s(-x_{ 1,0 }x_{ 0,2 }+x_{ 0,0 }x_{ 1,2 })+ t(-y_{ 1,1 }y_{ 0,3 }+y_{ 0,1 }y_{ 1,3 }) &= 0\\ s(x_{ 1,2 }x_{ 0,3 }-x_{ 0,2 }x_{ 1,3 }) + t(-y_{ 1,0 }y_{ 0,1 }+y_{ 0,0 }y_{ 1,1}) & = 0. \end{align*}
Away from the poles, the ideal is isomorphic to the spinor variety $S_{10} = OG_{+}(5,10)$ and a generic linear section (of dimension 1) is a canonical genus 7 curve as observed by Mukai. Over the poles $[s:t] = [1:0], [0:1]$, the fiber is a degeneration of $OG_{+}(5,10)$ and the generic linear section appears to be a tetragonal genus 7 curve. This description seems to be different from what Mukai gives. Are there nice ways to think about this degeneration? Perhaps from the theory of the moduli space of curves?
This degeneration appears in considering the variety of all possible (partial or full) topological twists of four-dimensional $\mathcal{N=4}$ super Yang--Mills theory [arXiv:1807.03766] where the $P^1$ is identified with the space of $GL$-twists considered by Kapustin and Witten.
