  Disordered hyperuniform materials are an emerging class of exotic amorphous
states of matter that endow them with singular physical properties. Here, we
generalize the Fourier-space based numerical construction procedure for
designing {\it isotropic} disordered hyperuniform two-phase heterogeneous
materials (i.e., composites) developed by Chen and Torquato [Acta Mater. {\bf
142}, 152 (2018)] to {\it anisotropic} microstructures by explicitly
incorporating the {\it vector-dependent} spectral density function ${\tilde
\chi}_{_V}({\bf k})$ of {\it arbitrary form} that is realizable. We demonstrate
the utility of the procedure by generating a wide spectrum of {\it anisotropic}
stealthy hyperuniform (SHU) microstructures with ${\tilde \chi}_{_V}({\bf k}) =
0$ for ${\bf k} \in \Omega$. We show how different exclusion-region shapes with
various discrete symmetries and varying size affect the resulting statistically
anisotropic microstructures as a function of the and phase volume fraction. We
find that, among other properties, the directional hyperuniform behaviors
imposed by the shape asymmetry (or anisotropy) of certain exclusion regions
give rise to distinct anisotropic structures and degree of uniformity in the
distribution of the phases on intermediate and large length scales along
different directions. Moreover, while the anisotropic exclusion regions impose
strong constraints on the {\it global} symmetry of the resulting media, they
can still possess almost isotropic {\it local} structures. Our construction
algorithm enables one to control the statistical anisotropy of composite
microstructures which is crucial to engineering directional optical, transport
and mechanical properties of two-phase composite media.
