We present a novel population-based local search algorithm for the {\em median genome problem}. The primary result of this article is that this probabilistic approach significantly improves the performance of ancestral genome reconstruction compared to existing methods, making it possible to tackle problems where the contemporary genomes may contain many hundreds of markers. Moreover, our method is not limited to triples of genomes, and thus solves the median genome problem in its generality. We show that in real application cases the computational results are highly robust, suggesting that we can interpret the computed median genomes as candidates carrying the semantics of ancestral architectures.