  We present a family of quantum error-correcting codes that support a
universal set of transversal logic gates using only local operations on a
two-dimensional array of physical qubits. The construction is a subsystem
version of color codes where gauge fixing through local measurements
dynamically determines which gates are transversal. Although the operations are
local, the underlying code is not topological in structure, which is how the
construction circumvents no-go constraints imposed by the Bravyi-K\"onig and
Pastawski-Yoshida theorems. We provide strong evidence that the encoding has no
error threshold in the conventional sense, though it is still possible to have
logical gates with error probability much lower than that of physical gates.
