  Given a pair of $\mathbb{Z}_2$-harmonic spinors (resp. 1-forms) on closed
Riemannian 3-manifolds $(Y_1, g_1)$ and $(Y_2,g_2)$, we construct
$\mathbb{Z}_2$-harmonic spinors (resp. 1-forms) on the connected sum $Y_1 \#
Y_2$ and the torus sum $Y_1 \cup_{T^2} Y_2$ using a gluing argument. The main
tool in the proof is a parameterized version of the Nash-Moser implicit
function theorem established by Donaldson and the second author. We use these
results to construct an abundance of new examples of $\mathbb Z_2$-harmonic
spinors and 1-forms. In particular, we prove that for every closed 3-manifold
$Y$, there exist infinitely many $\mathbb{Z}_2$-harmonic spinors with singular
sets representing infinitely many distinct isotopy classes of embedded links,
strengthening an existence theorem of Doan-Walpuski. Moreover, combining this
with previous results, our construction implies that if $b_1(Y) > 0$, there
exist infinitely many $\mathrm{spin}^c$ structures on $Y$ such that the moduli
space of solutions to the two-spinor Seiberg-Witten equations is non-empty and
non-compact.
