  Intrinsic alignment (IA) modelling and photometric redshift estimation are
two of the main sources of systematic uncertainty in weak lensing surveys. We
investigate the impact of redshift errors and their interplay with different IA
models. Generally, errors on the mean $\delta_z$ and on the width $\sigma_z$ of
the redshift bins can both lead to biases in cosmological constraints. We find
that such biases can, however, only be partially resolved by marginalizing over
$\delta_z$ and $\sigma_z$. For Stage-III surveys, $\delta_z$ and $\sigma_z$
cannot be well constrained due to limited statistics. The resulting biases are
thus sensitive to prior volume effects. For Stage-IV surveys, we observe that
marginalizing over the redshift parameters has an impact and reduces the bias.
We derive requirements on the uncertainty of $\sigma_z$ and $\delta_z$ for both
Stage-III and Stage-IV surveys. We assume that the redshift systematic errors
on $S_8$ should be less than half of the statistical errors, and the median
bias should be smaller than $0.25\sigma$. We find that the uncertainty on
$\delta_z$ has to be $\lesssim0.025$ for the NLA IA model with a Stage-III
survey. For $\sigma_z$, the requirement is met even for large uncertainties
$\leq0.3$. For the TATT IA model, the uncertainty on $\delta_z$ has to be
$\lesssim0.02$ and the uncertainty on $\sigma_z$ has to be $\lesssim0.2$. For
Stage-IV surveys, the uncertainty on $\delta_z$ has to be $\lesssim0.005$ and
the uncertainty on $\sigma_z$ should be $\lesssim0.1$, with no significant
dependence on the IA model. This required high precision will be a challenge
for the redshift calibration of these future surveys. Finally, we investigate
whether the interplay between redshift systematics and IA modelling can explain
the $S_8$-tension between cosmic shear results and CMB measurements. We find
that this is unlikely to explain the current $S_8$-tension.
