  In this work, we introduce an anyon condensation web that interconnects a
broad class of 2D fracton gauge theories with multipolar conservation laws at a
microscopic level. We find that condensation of anyons triggers the emergence
of additional spatially modulated symmetries, which has the general effect of
increasing the number of super-selection anyon sectors, as well as the ground
state degeneracy for systems with periodic boundary conditions. As explicit
examples, we start with the rank-2 toric code model and implement various anyon
condensation protocols, resulting in a range of 2D fracton-like theories, each
with a distinct gauge structure. We also expand the scope of anyon condensation
by introducing lattice defects into spatially modulated gauge theories and
demonstrate that these geometric defects can be viewed as effective anyon
condensations along the branch cut. Furthermore, we introduce the Multipartite
Entanglement Mutual Information measure as a diagnostic tool to differentiate
among various distinct multipole conserving phases. A captivating observation
is the UV sensitivity of the mutual information sourced from multipartite
entanglement in such spatially modulated gauge theories, which depends on the
geometric cut and the system size, and exhibits periodic oscillations at large
distances.
