  We introduce a method to construct $G_2$-instantons over compact
$G_2$-manifolds arising as the twisted connected sum of a matching pair of
building blocks [Kov03,KL11,CHNP12]. Our construction is based on gluing
$G_2$-instantons obtained from holomorphic bundles over the building blocks via
the first named author's work [SE11]. We require natural compatibility and
transversality conditions which can be interpreted in terms of certain
Lagrangian subspaces of a moduli space of stable bundles on a K3 surface.
