We propose a method to substantially increase the flexibility and power of template fitting-based photometric redshifts by transforming a large number of galaxy spectral templates into a corresponding collection of 'fuzzy archetypes' using a suitable set of perturbative priors designed to account for empirical variation in dust attenuation and emission-line strengths. To bypass widely separated degeneracies in parameter space (e.g. the redshift-reddening degeneracy), we train self-organizing maps (SOMs) on large 'model catalogues' generated from Monte Carlo sampling of our fuzzy archetypes to cluster the predicted observables in a topologically smooth fashion. Subsequent sampling over the SOM then allows full reconstruction of the relevant probability distribution functions (PDFs). This combined approach enables the multimodal exploration of known variation among galaxy spectral energy distributions with minimal modelling assumptions. We demonstrate the power of this approach to recover full redshift PDFs using discrete Markov chain Monte Carlo sampling methods combined with SOMs constructed from Large Synoptic Survey Telescope ugrizY and Euclid YJH mock photometry.