We study modular invariance of normalized supercharacters of tame integrable modules over an affine Lie superalgebra, associated to an arbitrary basic Lie superalgebra $\mathfrak g$. For this we develop a several step modification process of multivariable mock theta functions, where at each step a Zwegers' type ‘modifier’ is used. We show that the span of the resulting modified normalized supercharacters is $\operatorname{SL}_2(\mathbb Z)$-invariant, with the transformation matrix equal, in the case the Killing form on $\mathfrak g$ is non-degenerate, to that for the basic defect 0 subalgebra $\mathfrak g^!$ of $\mathfrak g$, orthogonal to a maximal isotropic set of roots of $\mathfrak g$.