  The Twin Higgs model seeks to address the little hierarchy problem by making
the Higgs a pseudo-Goldstone of a global $SU(4)$ symmetry that is spontaneously
broken to $SU(3)$. Gauge and Yukawa couplings, which explicitly break $SU(4)$,
enjoy a discrete $\mathbb{Z}_2$ symmetry that accidentally maintains $SU(4)$ at
the quadratic level and therefore keeps the Higgs light. Contrary to most
beyond the Standard Model theories, the quadratically divergent corrections to
the Higgs mass are cancelled by a mirror sector, which is uncharged under the
Standard Model groups. However, the Twin Higgs with an exact $\mathbb{Z}_2$
symmetry leads to equal vevs in the Standard Model and mirror sectors, which is
phenomenologically unviable. An explicit $\mathbb{Z}_2$ breaking potential must
then be introduced and tuned against the $SU(4)$ breaking terms to produce a
hierarchy of vevs between the two sectors. This leads to a moderate but
non-negligible tuning. We propose a model to alleviate this tuning, without the
need for an explicit $\mathbb{Z}_2$ breaking sector. The model consists of two
$SU(4)$ fundamental Higgses, one whose vacuum preserves $\mathbb{Z}_2$ and one
whose vacuum breaks it. As the interactions between the two Higgses are turned
on, the $\mathbb{Z}_2$ breaking is transmitted from the broken to the unbroken
sector and a small hierarchy of vevs is naturally produced. The presence of an
effective tadpole and feedback between the two Higgses lead to a sizable
improvement of the tuning. The resulting Higgs boson is naturally very Standard
Model like.
