  We present a field-based signal extraction of weak lensing from noisy
observations on the curved and masked sky. We test the analysis on a simulated
Euclid-like survey, using a Euclid-like mask and noise level. To make optimal
use of the information available in such a galaxy survey, we present a Bayesian
method for inferring the angular power spectra of the weak lensing fields,
together with an inference of the noise-cleaned tomographic weak lensing shear
and convergence (projected mass) maps. The latter can be used for field-level
inference with the aim of extracting cosmological parameter information
including non-gaussianity of cosmic fields. We jointly infer all-sky $E$-mode
and $B$-mode tomographic auto- and cross-power spectra from the masked sky, and
potentially parity-violating $EB$-mode power spectra, up to a maximum multipole
of $\ell_{\rm max}=2048$. We use Hamiltonian Monte Carlo sampling, inferring
simultaneously the power spectra and denoised maps with a total of $\sim 16.8$
million free parameters. The main output and natural outcome is the set of
samples of the posterior, which does not suffer from leakage of power from $E$
to $B$ unless reduced to point estimates. However, such point estimates of the
power spectra, the mean and most likely maps, and their variances and
covariances, can be computed if desired.
