Recently attention has been turned to the problem of reconstructing complete ancestral sequences from large multiple alignments. Successful generation of these genome-wide reconstructions will facilitate a greater knowledge of the events that have driven evolution. We present a new evolutionary alignment modeler, called “Ortheus,” for inferring the evolutionary history of a multiple alignment, in terms of both substitutions and, importantly, insertions and deletions. Based on a multiple sequence probabilistic transducer model of the type proposed by Holmes, Ortheus uses efficient stochastic graph-based dynamic programming methods. Unlike other methods, Ortheus does not rely on a single fixed alignment from which to work. Ortheus is also more scaleable than previous methods while being fast, stable, and open source. Large-scale simulations show that Ortheus performs close to optimally on a deep mammalian phylogeny. Simulations also indicate that significant proportions of errors due to insertions and deletions can be avoided by not assuming a fixed alignment. We additionally use a challenging hold-out cross-validation procedure to test the method; using the reconstructions to predict extant sequence bases, we demonstrate significant improvements over using closest extant neighbor sequences. Accompanying this paper, a new, public, and genome-wide set of Ortheus ancestor alignments provide an intriguing new resource for evolutionary studies in mammals. As a first piece of analysis, we attempt to recover “fossilized” ancestral pseudogenes. We confidently find 31 cases in which the ancestral sequence had a more complete sequence than any of the extant sequences.