Phylogenies depicting the evolutionary history of genetically heterogeneous subpopulations of cells from the same cancer, i.e., cancer phylogenies, offer valuable insights about cancer development and guide treatment strategies. Many methods exist that reconstruct cancer phylogenies using point mutations detected with bulk DNA sequencing. However, these methods become inaccurate when reconstructing phylogenies with more than 30 mutations, or, in some cases, fail to recover a phylogeny altogether. Here, we introduce Orchard, a cancer phylogeny reconstruction algorithm that is fast and accurate using up to 1000 mutations. Orchard samples without replacement from a factorized approximation of the posterior distribution over phylogenies, a novel result derived in this paper. Each factor in this approximate posterior corresponds to a conditional distribution for adding a new mutation to a partially built phylogeny. Orchard optimizes each factor sequentially, generating a sequence of incrementally larger phylogenies that ultimately culminate in a complete tree containing all mutations. Our evaluations demonstrate that Orchard outperforms state-of-the-art cancer phylogeny reconstruction methods in reconstructing more plausible phylogenies across 90 simulated cancers and 14 B-progenitor acute lymphoblastic leukemias (B-ALLs). Remarkably, Orchard accurately reconstructs cancer phylogenies using up to 1,000 mutations. Additionally, we demonstrate that the large and accurate phylogenies reconstructed by Orchard are useful for identifying patterns of somatic mutations and genetic variations among distinct cancer cell subpopulations.