We prove the existence of singular harmonic $\mathbf{Z}_2$ spinors on $3$‑manifolds with $b_1 \gt 1$. The proof relies on a wall-crossing formula for solutions to the Seiberg–Witten equation with two spinors. The existence of singular harmonic $\mathbf{Z}_2$ spinors and the shape of our wall-crossing formula shed new light on recent observations made by Joyce [Joy17] regarding Donaldson and Segal’s proposal for counting $G_2$-instantons [DS11].