  A set of $2n$ points in $\mathbb{P}^{n-1}$ is self-dual if it is invariant
under the Gale transform. Motivated by Mukai's work on canonical curves,
Petrakiev showed that a general self-dual set of $14$ points in $\mathbb{P}^6$
arises as the intersection of the Grassmannian ${\rm Gr}(2,6)$ in its Pl\"ucker
embedding in $\mathbb{P}^{14}$ with a linear space of dimension $6$. In this
paper we focus on the inverse problem of recovering such a linear space
associated to a general self-dual set of points. We use numerical homotopy
continuation to approach the problem and implement an algorithm in Julia to
solve it. Along the way we also implement the forward problem of slicing
Grassmannians and use it to experimentally study the real solutions to this
problem.
