  We study the conformal bootstrap for systems of correlators involving
non-identical operators. The constraints of crossing symmetry and unitarity for
such mixed correlators can be phrased in the language of semidefinite
programming. We apply this formalism to the simplest system of mixed
correlators in 3D CFTs with a $\mathbb{Z}_2$ global symmetry. For the leading
$\mathbb{Z}_2$-odd operator $\sigma$ and $\mathbb{Z}_2$-even operator
$\epsilon$, we obtain numerical constraints on the allowed dimensions
$(\Delta_\sigma, \Delta_\epsilon)$ assuming that $\sigma$ and $\epsilon$ are
the only relevant scalars in the theory. These constraints yield a small closed
region in $(\Delta_\sigma, \Delta_\epsilon)$ space compatible with the known
values in the 3D Ising CFT.
