Intra-tumour genetic heterogeneity is the result of ongoing evolutionary change within each cancer. The expansion of genetically distinct sub-clonal populations may explain the emergence of drug resistance, and if so, would have prognostic and predictive utility. However, methods for objectively quantifying tumour heterogeneity have been missing and are particularly difficult to establish in cancers where predominant copy number variation prevents accurate phylogenetic reconstruction owing to horizontal dependencies caused by long and cascading genomic rearrangements. To address these challenges, we present MEDICC, a method for phylogenetic reconstruction and heterogeneity quantification based on a Minimum Event Distance for Intra-tumour Copy-number Comparisons. Using a transducer-based pairwise comparison function, we determine optimal phasing of major and minor alleles, as well as evolutionary distances between samples, and are able to reconstruct ancestral genomes. Rigorous simulations and an extensive clinical study show the power of our method, which outperforms state-of-the-art competitors in reconstruction accuracy, and additionally allows unbiased numerical quantification of tumour heterogeneity. Accurate quantification and evolutionary inference are essential to understand the functional consequences of tumour heterogeneity. The MEDICC algorithms are independent of the experimental techniques used and are applicable to both next-generation sequencing and array CGH data.