  Given a $\mathbb Z_2$-harmonic spinor satisfying some genericity assumptions,
this article constructs a 1-parameter family of two-spinor Seiberg-Witten
monopoles converging to it after renormalization. The proof is a gluing
construction beginning with model solutions on a neighborhood of the $\mathbb
Z_2$-harmonic spinor's singular set. The gluing is complicated by the presence
of an infinite-dimensional obstruction bundle for the singular limiting
linearized operator. This difficulty is overcome by introducing a
generalization of Donaldson's alternating method in which a deformation of the
$\mathbb Z_2$-harmonic spinor's singular set is chosen at each stage of the
alternating iteration to cancel the obstruction components.
