The classification of the minor bodies of the Solar System based on observables has been continuously developed and iterated over the past 40 yr. While prior iterations followed either the availability of large observational campaigns or new instrumental capabilities opening new observational dimensions, we see the opportunity to improve primarily upon the established methodology. We developed an iteration of the asteroid taxonomy which allows the classification of partial and complete observations (i.e. visible, near-infrared, and visible-near-infrared spectrometry) and which reintroduces the visual albedo into the classification observables. The resulting class assignments are given probabilistically, enabling the uncertainty of a classification to be quantified. We built the taxonomy based on 2983 observations of 2125 individual asteroids, representing an almost tenfold increase of sample size compared with the previous taxonomy. The asteroid classes are identified in a lower-dimensional representation of the observations using a mixture of common factor analysers model. We identify 17 classes split into the three complexes C, M, and S, including the new Z-class for extremely-red objects in the main belt. The visual albedo information resolves the spectral degeneracy of the X-complex and establishes the P-class as part of the C-complex. We present a classification tool which computes probabilistic class assignments within this taxonomic scheme from asteroid observations, intrinsically accounting for degeneracies between classes based on the observed wavelength region. The taxonomic classifications of 6038 observations of 4526 individual asteroids are published. The ability to classify partial observations and the reintroduction of the visual albedo into the classification provide a taxonomy which is well suited for the current and future datasets of asteroid observations, in particular provided by the Gaia, MITHNEOS, NEO Surveyor, and SPHEREx surveys.