Bacon-Shor codes are quantum subsystem codes which are constructed by combining together two quantum repetition codes, one protecting against Z (phase) errors and the other protecting against X (bit flip) errors. In many situations, for example flux qubits, the noise is biased such that Z errors are much more common than X errors; in these cases it is natural to consider an asymmetric Bacon-Shor code where the code protecting against Z errors is longer than the code protecting against X errors. This work provides fault-tolerant gadget constructions to achieve universal fault-tolerant quantum computation using asymmetric Bacon-Shor codes and controlled-Z gates as the only two-qubit gates. The qubits can be arranged in a constant-width ribbon and all gates performed on single qubits or neighboring pairs of qubits. In the presence of biased noise, these constructions allow for powerful reductions in the error rate with modest resource overhead.