We present a proof of principle for a new kind of stepwise algorithm for unrooted binary gene-order phylogenies. This method incorporates a simple look-ahead inspired by Dollo’s law, while simultaneously reconstructing each ancestor (sometimes referred to as hypothetical taxonomic units “HTU”). We first present a generic version of the algorithm illustrating a necessary consequence of Dollo characters. In a concrete application we use generalized oriented gene adjacencies and maximum weight matching (MWM) to reconstruct fragments of monoploid ancestral genomes as HTUs. This is applied to three flowering plant orders while estimating phylogenies for these orders in the process. We discuss how to improve on the extensive computing times that would be necessary for this method to handle larger trees.