Inference in cosmology often starts with noisy observations of random fields on the celestial sphere, such as maps of the microwave background radiation, continuous maps of cosmic structure in different wavelengths, or maps of point tracers of the cosmological fields. Almanac uses Hamiltonian Monte Carlo sampling to infer the underlying all-sky noiseless maps of cosmic structures, in multiple redshift bins, together with their auto- and cross-power spectra. It can sample many millions of parameters, handling the highly variable signal-to-noise of typical cosmological signals, and it provides science-ready posterior data products. In the case of spin-weight 2 fields, Almanac infers E- and B-mode power spectra and parity-violating EB power, and, by sampling the full posteriors rather than point estimates, it avoids the problem of EB-leakage. For theories with no B-mode signal, inferred non-zero B-mode power may be a useful diagnostic of systematic errors or an indication of new physics. Almanac's aim is to characterise the statistical properties of the maps, with outputs that are completely independent of the cosmological model, beyond an assumption of statistical isotropy. Inference of parameters of any particular cosmological model follows in a separate analysis stage. We demonstrate our signal extraction on a CMB-like experiment.