Consider strong approximation for algebraic varieties defined over a number field k k . Let S S be a finite set of places of k k containing all archimedean places. Let E E be an elliptic curve of positive Mordell–Weil rank and let A A be an abelian variety of positive dimension and of finite Mordell–Weil group. For an arbitrary finite set T \mathfrak {T} of torsion points of E × A E\times A , denote by X X its complement. Supposing the finiteness of ⨿ ⨿ ( E × A ) {\amalg \kern -.25pc\amalg } (E\times A) , we prove that X X satisfies strong approximation with Brauer–Manin obstruction off S S if and only if the projection of T \mathfrak {T} to A A contains no k k -rational points.