  The fractionalization of global symmetry charges is a striking hallmark of
topological quantum order. Here, we discuss the fractionalization of subsystem
symmetries in two-dimensional topological phases. In line with previous no-go
arguments, we show that subsystem symmetry fractionalization is not possible in
many cases due to the additional rigid geometric structure of the symmetries.
However, we identify a new mechanism that allows fractionalization, involving
global relations between macroscopically many symmetry generators. We find that
anyons can fractionalize such relations, meaning that the total charge carried
under all generators involved in the global relation is non-trivial, despite
the fact that these generators multiply to the identity. We first discuss the
general algebraic framework needed to characterize this new type of
fractionalization, and then explore this framework using a number of exactly
solvable models with $\mathbb{Z}_2$ topological order, including models having
line and fractal symmetries. These models all showcase another necessary
property of subsystem symmetry fractionalization: fractionalized anyons must
have restricted mobility when the symmetry is enforced, such that they are
confined to a single line or point in the case of line and fractal symmetries,
respectively. Looking forward, we expect that our identification of the
importance of global relations in fractionalization will hold significance for
the classification of phases with subsystem symmetries in all dimensions.
