  Fault-tolerant cluster states form the basis for scalable measurement-based
quantum computation. Recently, new stabilizer codes for scalable circuit-based
quantum computation have been introduced that have very high thresholds under
biased noise where the qubit predominantly suffers from one type of error, e.g.
dephasing. However, extending these advances in stabilizer codes to generate
high-threshold cluster states for biased noise has been a challenge, as the
standard method for foliating stabilizer codes to generate fault-tolerant
cluster states does not preserve the noise bias. In this work, we overcome this
barrier by introducing a generalization of the cluster state that allows us to
foliate stabilizer codes in a bias-preserving way. As an example of our
approach, we construct a foliated version of the XZZX code which we call the
XZZX cluster state. We demonstrate that under a circuit-level noise model, our
XZZX cluster state has a threshold more than double the usual cluster state
when dephasing errors are more likely than errors which cause bit flips by a
factor of $O(100)$ or more.
