  We determine the solution to the classical master equation for a 3D
topological field theory with Wess-Zumino term and an underlying geometrical
structure of a twisted R-Poisson manifold on its target space. The graded
geometry of the target space departs from the usual QP structure encountered in
the AKSZ construction of topological sigma models, the obstruction being
attributed to the presence of the Wess-Zumino 4-form. Due to the
inapplicability of the AKSZ construction in this case, we set up the
traditional BV/BRST formalism for twisted R-Poisson sigma models in any
dimension, which feature an open gauge algebra and constitute multiple stages
reducible constrained Hamiltonian systems. An unusual feature of the theories
is that it exhibits non-linear openness of the gauge algebra, in other words
products of the equations of motion appear in it. Nevertheless, we find the BV
action in presence of the 4-form twist in 3D, namely for a specific 4-form
twisted (pre-)Courant sigma model. Moreover, we provide a complete set of
explicit formulas for the off-shell nilpotent BV operator for untwisted
R-Poisson sigma models in any dimension.
