Structural molecular biology depends crucially on computational techniques that compare protein three-dimensional structures and generate structural alignments (the assignment of one-to-one correspondences between subsets of amino acids based on atomic coordinates.) Despite its importance, the structural alignment problem has not been formulated, much less solved, in a consistent and reliable way. To overcome these difficulties, we present here a framework for precise inference of structural alignments, built on the Bayesian and information-theoretic principle of Minimum Message Length (MML). The quality of any alignment is measured by its explanatory power -- the amount of lossless compression achieved to explain the protein coordinates using that alignment. We have implemented this approach in the program MMLigner (http://lcb.infotech.monash.edu.au/mmligner) to distinguish statistically significant alignments, not available elsewhere. We also demonstrate the reliability of MMLigner's alignment results compared with the state of the art. Importantly, MMLigner can also discover different structural alignments of comparable quality, a challenging problem for oligomers and protein complexes.