  This paper is concerned with the explicit computation of the limiting
distribution function of the largest real eigenvalue in the real Ginibre
ensemble when each real eigenvalue has been removed independently with constant
likelihood. We show that the recently discovered integrable structures in
\cite{BB} generalize from the real Ginibre ensemble to its thinned equivalent.
Concretely, we express the aforementioned limiting distribution function as a
convex combination of two simple Fredholm determinants and connect the same
function to the inverse scattering theory of the Zakharov-Shabat system. As
corollaries, we provide a Zakharov-Shabat evaluation of the ensemble's real
eigenvalue generating function and obtain precise control over the limiting
distribution function's tails. The latter part includes the explicit
computation of the usually difficult constant factors.
