  Cosmic shear data contains a large amount of cosmological information
encapsulated in the non-Gaussian features of the weak lensing mass maps. This
information can be extracted using non-Gaussian statistics. We compare the
constraining power in the $\Omega_{\mathrm{m}} - \sigma_8$ plane of three
map-based non-Gaussian statistics with the angular power spectrum, namely;
peak/minimum counts and Minkowski functionals. We further analyze the impact of
tomography and systematic effects originating from galaxy intrinsic alignments,
multiplicative shear bias and photometric redshift systematics. We forecast the
performance of the statistics for a stage-3-like weak lensing survey and
restrict ourselves to scales $\geq$ 10 arcmin. We find, that in our setup, the
considered non-Gaussian statistics provide tighter constraints than the angular
power spectrum. The peak counts show the greatest potential, increasing the
Figure-of-Merit (FoM) in the $\Omega_{\mathrm{m}} - \sigma_8$ plane by a factor
of about 4. A combined analysis using all non-Gaussian statistics in addition
to the power spectrum increases the FoM by a factor of 5 and reduces the error
on $S_8$ by $\approx$ 25\%. We find that the importance of tomography is
diminished when combining non-Gaussian statistics with the angular power
spectrum. The non-Gaussian statistics indeed profit less from tomography and
the minimum counts and Minkowski functionals add some robustness against galaxy
intrinsic alignment in a non-tomographic setting. We further find that a
combination of the angular power spectrum and the non-Gaussian statistics
allows us to apply conservative scale cuts in the analysis, thus helping to
minimize the impact of baryonic and relativistic effects, while conserving the
cosmological constraining power. We make the code that was used to conduct this
analysis publicly available.
