  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 $ 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 subalgebra $ \mathfrak{g}^! $ of $
\mathfrak{g}, $ orthogonal to a maximal isotropic set of roots of $
\mathfrak{g}. $
