QCD from chirally segregated, gauged $SU(3)_L \times SU(3)_R$? There are already theory papers out there in which color $SU(3)_C$ is actually the diagonal subgroup of multiple $SU(3)$ factors. But due to a comment by @zooby, a new twist on this idea occurred to me: what if one $SU(3)$ couples to left-handed quarks only, and the other $SU(3)$ couples to right-handed quarks only? (This is what I mean by "chirally segregated".) The point is that the Higgs-mediated Yukawa coupling of these Weyl-fermion quarks to each other, will still create de-facto Dirac-fermion quarks; and those "Dirac quarks" might, under certain circumstances, couple to the diagonal $SU(3)$. 
So I'm interested in (1) theoretical insight regarding how such entities would actually behave (2) if their behavior does resemble reality, how we could look for differences from standard model (e.g. top quark decay?). 
 A: The pathological part is that you are asking for a gauged $SU(3)_L\times SU(3)_R$ group. The 8 axial (L-R) generators of the symmetry (not closing to any Lie algebra!) would be explicitly broken by the quark mass term, since this would only be invariant under the diagonal ("color") SU(3), the remaining 8 vector (L+R) generators. So the gauge couplings, etc.. of these 8 axial generators would be inconsistent, unrenormalizable, etc... the standard pathologies of non-gauge theories. The theory would be meaningless, so worse than unrealistic.
The Higgs Yukawa couplings inducing fermion Dirac masses are very different: They are invariant under $SU(2)_L$ by dint of saturation to the Higgs L-doublet, and there is no $SU(2)_R$, gauged or ungauged, to speak of.
However, if you forfeited the gauging (really, what do you need it for?), 
then you are already describing the real light flavor world: the chiral $SU(3)_L\times SU(3)_R$ flavor symmetry of the light quarks, u,d,s, global and (slightly) explicitly broken by the mass terms--that is to say, the masses of these quarks are "smaller" than QCD scales (Λ, whatever). 
If we ignored these mass terms, this symmetry would be good. On top of this, QCD does break it (the 8 axials again) dynamically (~spontaneously, so it just hides it) and leaves remnants around, pseudogoldstone boson mesons, (π, η, K, an eightfold-way flavor octet.). The surviving diagonal subgroup is the celebrated vector flavor SU(3), Gell-Mann's "eightfold way". It is explicitly, slightly broken by the quark masses, which also break the 8 axials explicitly, so the pseudoscalar mesons are "slightly" massive (their masses being abnormally light, on the QCD scale). All this is real, useful, and significant, and is all made possible because the chiral group was not gauged. Explicit breaking of global symmetries is safe. (Hell, even global flavor chiral anomalies are safe, and drive low energy physics, the πππKK coupling, the πγγ coupling, etc.)


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*Full disclosure footnote: Of course, everything under the sun has been speculated about, so, in 1987, Frampton and Glashow have asked a broad question about your chiral color, leaving its spontaneous/dynamical breaking and causes up in the air and for model-builders to provide plausible-looking caricatures for it. Hokey combinations of Higgses, unrenormalizable interactions,  and further ultra-strong gauge interactions could get something close to this vision, but, to my knowledge, all extant attempts have failed at multiple levels.

