  We employ the formalism developed in \cite{Mentasti:2023gmg} and
\cite{Bartolo_2022} to study the prospect of detecting an anisotropic
Stochastic Gravitational Wave Background (SGWB) with the Laser Interferometer
Space Antenna (LISA) alone, and combined with the proposed space-based
interferometer Taiji. Previous analyses have been performed in the frequency
domain only. Here, we study the detectability of the individual coefficients of
the expansion of the SGWB in spherical harmonics, by taking into account the
specific motion of the satellites. This requires the use of time-dependent
response functions, which we include in our analysis to obtain an optimal
estimate of the anisotropic signal. We focus on two applications. Firstly, the
reconstruction of the anisotropic galactic signal without assuming any prior
knowledge of its spatial distribution. We find that both LISA and LISA with
Taiji cannot put tight constraints on the harmonic coefficients for realistic
models of the galactic SGWB. We then focus on the discrimination between a
galactic signal of known morphology but unknown overall amplitude and an
isotropic extragalactic SGWB component of astrophysical origin. In this case,
we find that the two surveys can confirm, at a confidence level $\gtrsim
3\sigma$, the existence of both the galactic and extragalactic background if
both have amplitudes as predicted in standard models. We also find that, in the
LISA-only case, the analysis in the frequency domain (under the assumption of a
time average of data taken homogeneously across the year) provides a nearly
identical determination of the two amplitudes as compared to the optimal
analysis.
