  An abstract system of congruences describes a way of partitioning a space
into finitely many pieces satisfying certain congruence relations. Examples of
abstract systems of congruences include paradoxical decompositions and
$n$-divisibility of actions. We consider the general question of when there are
realizations of abstract systems of congruences satisfying various
measurability constraints. We completely characterize which abstract systems of
congruences can be realized by nonmeager Baire measurable pieces of the sphere
under the action of rotations on the $2$-sphere. This answers a question of
Wagon. We also construct Borel realizations of abstract systems of congruences
for the action of $\mathsf{PSL}_2(\mathbb{Z})$ on $\mathsf{P}^1(\mathbb{R})$.
The combinatorial underpinnings of our proof are certain types of decomposition
of Borel graphs into paths. We also use these decompositions to obtain some
results about measurable unfriendly colorings.
