  The analytical study of confinement in lattice gauge theories (LGTs) remains
a difficult task to this day. Taking a geometric perspective on confinement, we
develop a real-space renormalization group (RG) formalism for $\mathbb{Z}_2$
LGTs using percolation probability as a confinement order parameter. The RG
flow we analyze is constituted by both the percolation probability and the
coupling parameters. We consider a classical $\mathbb{Z}_2$ LGT in two
dimensions, with matter and thermal fluctuations, and analytically derive the
confinement phase diagram. We find good agreement with numerical and exact
benchmark results and confirm that a finite matter density enforces confinement
at $T<\infty$ in the model we consider. Our RG scheme enables future analytical
studies of $\mathbb{Z}_2$ LGTs with matter and quantum fluctuations and beyond.
