  A Lagrangian BRST quantization for generic gauge theories in general
irreducible non-Abelian hypergauges is proposed on a basis of the multilevel
Batalin--Tyutin formalism and a special BV--BFV dual description for a
reducible gauge model in a symplectic supermanifold $\mathcal{M}_0$ locally
parameterized by antifields for Lagrangian multipliers and by the fields of the
BV method. The quantization rules are based on a set of nilpotent anticommuting
operators $\Delta^\mathcal{M}, {V}^\mathcal{M}, {U}^\mathcal{M}$ defined using
some odd and even symplectic structures in a supersymplectic manifold
$\mathcal{M}$ whose local representation is an odd (co)tangent bundle over
$\mathcal{M}_0$ provided by the choice of a flat Fedosov connection and a
compatible non-symplectic metric in $\mathcal{M}_0$. The generating functional
of Green's functions is constructed in terms of general coordinates in
$\mathcal{M}$ with the help of contracting homotopy operators with respect to
${V}^\mathcal{M}$ and ${U}^\mathcal{M}$. We prove the gauge independence of the
S-matrix and derive the Ward identity.
