  A large class of type-I fracton models, including the X-cube model, have been
found to be fixed points of the foliated renormalization group (RG). The system
size of such foliated models can be changed by adding or removing decoupled
layers of $2$D topological states and continuous deformation of the
Hamiltonian. In this paper, we study a closely related model -- the Ising
cage-net model -- and find that this model is not foliated in the same sense.
In fact, we point out certain unnatural restrictions in the foliated RG, and
find that removing these restrictions leads to a generalized foliated RG under
which the Ising cage-net model is a fixed point, and which includes the
original foliated RG as a special case. The Ising cage-net model thus gives a
prototypical example of the generalized foliated RG, and its system size can be
changed either by condensing / uncondensing bosonic planon excitations near a
2D plane or through a linear depth quantum circuit in the same plane. We show
that these two apparently different RG procedures are closely related, as they
lead to the same gapped boundary when implemented in part of a plane. Finally,
we briefly discuss the implications for foliated fracton phases, whose
universal properties will need to be reexamined in light of the generalized
foliated RG.
