  Let $L\subset X$ be a not necessarily orientable relatively $Pin$ Lagrangian
submanifold in a symplectic manifold $X$. We construct a family of cyclic
unital curved $A_\infty$ structures on differential forms on $L$ with values in
the local system of graded non-commutative rings given by the tensor algebra of
the orientation local system of $L$. The family of $A_\infty$ structures is
parameterized by the cohomology of $X$ relative to $L$ and satisfies properties
analogous to the axioms of Gromov-Witten theory. On account of the
non-orientability of $L,$ the evaluation maps of moduli spaces of
$J$-holomorphic disks with boundary in $L$ may not be relatively orientable. To
deal with this problem, we use recent results on orientor calculus.
