Morphological data play a key role in the inference of biological relationships and evolutionary history and are essential for the interpretation of the fossil record. The hierarchical interdependence of many morphological characters, however, complicates phylogenetic analysis. In particular, many characters only apply to a subset of terminal taxa. The widely used “reductive coding” approach treats taxa in which a character is inapplicable as though the character’s state is simply missing (unknown). This approach has long been known to create spurious tree length estimates on certain topologies, potentially leading to erroneous results in phylogenetic searches—but pratical solutions have yet to be proposed and implemented. Here, we present a single-character algorithm for reconstructing ancestral states in reductively coded data sets, following the theoretical guideline of minimizing homoplasy over all characters. Our algorithm uses up to three traversals to score a tree, and a fourth to fully resolve final states at each node within the tree. We use explicit criteria to resolve ambiguity in applicable/inapplicable dichotomies, and to optimize missing data. So that it can be applied to single characters, the algorithm employs local optimization; as such, the method provides a fast but approximate inference of ancestral states and tree score. The application of our method to published morphological data sets indicates that, compared to traditional methods, it identifies different trees as “optimal.” As such, the use of our algorithm to handle inapplicable data may significantly alter the outcome of tree searches, modifying the inferred placement of living and fossil taxa and potentially leading to major differences in reconstructions of evolutionary history.