  Calibrating photometric redshift errors in weak lensing surveys with external
data is extremely challenging. We show that both Gaussian and outlier photo-z
parameters can be self-calibrated from the data alone. This comes at no cost
for the neutrino masses, curvature and dark energy equation of state $w_0$, but
with a 65% degradation when both $w_0$ and $w_a$ are varied. We perform a
realistic forecast for the Vera Rubin Observatory (VRO) Legacy Survey of Space
and Time (LSST) 3x2 analysis, combining cosmic shear, projected galaxy
clustering and galaxy - galaxy lensing. We confirm the importance of
marginalizing over photo-z outliers. We examine a subset of internal
cross-correlations, dubbed "null correlations", which are usually ignored in
3x2 analyses. Despite contributing only $\sim$ 10% of the total
signal-to-noise, these null correlations improve the constraints on photo-z
parameters by up to an order of magnitude. Using the same galaxy sample as
sources and lenses dramatically improves the photo-z uncertainties too.
Together, these methods add robustness to any claim of detected new Physics,
and reduce the statistical errors on cosmology by 15% and 10% respectively.
Finally, including CMB lensing from an experiment like Simons Observatory or
CMB-S4 improves the cosmological and photo-z posterior constraints by about
10%, and further improves the robustness to systematics. To give intuition on
the Fisher forecasts, we examine in detail several toy models that explain the
origin of the photo-z self-calibration. Our Fisher code LaSSI (Large-Scale
Structure Information), which includes the effect of Gaussian and outlier
photo-z, shear multiplicative bias, linear galaxy bias, and extensions to
$\Lambda$CDM, is publicly available at https://github.com/EmmanuelSchaan/LaSSI .
