  We consider a realisation of composite Higgs models in the context of $SU(6)/
SO(6)$ symmetry, which features a custodial bi-triplet, two Higgs doublets and
dark matter candidates. This model can arise from an underlying gauge-fermion
theory. The general vacuum structure is explored using the top partial
compositeness to generate a special vacuum characterised by a single angle
aligned with the first Higgs doublet. We present the CP and Dark Matter
$\mathbb{Z}_2$ parity in two different pNGB bases and analyse the spectra in
the absence of tadpoles and tachyons. For the phenomenology, we discuss the
constraints from electroweak precision tests and from a potentially light
CP-odd singlet (other than the Dark Matter) in the model.
