  We explicitly construct a class of holographic quantum error correction codes
with non-trivial centers in the code subalgebra. Specifically, we use the
Bacon-Shor codes and perfect tensors to construct a gauge code (or a stabilizer
code with gauge-fixing), which we call the holographic hybrid code. This code
admits a local log-depth encoding/decoding circuit, and can be represented as a
holographic tensor network which satisfies an analog of the Ryu-Takayanagi
formula and reproduces features of the sub-region duality. We then construct
approximate versions of the holographic hybrid codes by "skewing" the code
subspace, where the size of skewing is analogous to the size of the
gravitational constant in holography. These approximate hybrid codes are not
necessarily stabilizer codes, but they can be expressed as the superposition of
holographic tensor networks that are stabilizer codes. For such constructions,
different logical states, representing different bulk matter content, can
"back-react" on the emergent geometry, resembling a key feature of gravity. The
locality of the bulk degrees of freedom becomes subspace-dependent and
approximate. Such subspace-dependence is manifest from the point of view of the
"entanglement wedge" and bulk operator reconstruction from the boundary. Exact
complementary error correction breaks down for certain bipartition of the
boundary degrees of freedom; however, a limited, state-dependent form is
preserved for particular subspaces. We also construct an example where the
connected two-point correlation functions can have a power-law decay. Coupled
with known constraints from holography, a weakly back-reacting bulk also forces
these skewed tensor network models to the "large $N$ limit" where they are
built by concatenating a large $N$ number of copies.
