We update the field-level inference code KARMMA to enable tomographic forward-modelling of shear maps. Our code assumes a lognormal prior on the convergence field, and properly accounts for the cross-covariance in the lensing signal across tomographic source bins. We use mock weak lensing data from N-body simulations to validate our mass-mapping forward model by comparing our posterior maps to the input convergence fields. We find that KARMMA produces more accurate reconstructions than traditional mass-mapping algorithms. More-over, the KARMMA posteriors reproduce all statistical properties of the input density field we tested -- one- and two-point functions, and the peak and void number counts -- with accuracy. Our posteriors exhibit a small bias that increases with decreasing source redshift, but these biases are small compared to the statistical uncertainties of current (DES) cosmic shear surveys. Finally, we apply KARMMA to Dark Energy Survey Year 3 (DES-Y3) weak lensing data, and verify that the two point shear correlation function is well fit by the correlation function of the reconstructed convergence field. This is a non-trivial test that traditional mass mapping algorithms fail. The code is publicly available at https://github.com/Supranta/KaRMMa.git. KARMMA DES-Y3 mass maps are publicly available at https://zenodo.org/records/10672062.