  We present a flavour non-universal extension of the Standard Model combined
with the idea of Higgs compositeness. At the TeV scale, the gauge groups
$SU(2)_R$ and $U(1)_{B-L}$ are assumed to act in a non-universal manner on
light- and third-generation fermions, while the Higgs emerges as a pseudo
Nambu-Goldstone boson of the spontaneous global symmetry breaking $Sp(4)\to
SU(2)_L\times SU(2)_R^{[3]}$, attributed to new strong dynamics. The flavour
deconstruction means the couplings of the light families to the composite
sector (and therefore the pNGB Higgs) are suppressed by powers of a heavy mass
scale (from which the Higgs is nevertheless shielded by compositeness),
explaining the flavour puzzle. We present a detailed analysis of the
radiatively generated Higgs potential, showing how this intrinsically-flavoured
framework has the ingredients to justify the unavoidable tuning in the Higgs
potential necessary to separate electroweak and composite scales. This happens
for large enough values of the $SU(2)_R^{[3]}$ gauge coupling and light enough
flavoured gauge bosons resulting from the deconstruction, whose phenomenology
is also investigated. The model is compatible with current experimental bounds
and predicts new states at the TeV scale, which are within the reach of near
future experimental searches.
