The genomic median problem is an optimization problem inspired by a biological issue: it aims at finding the chromosome organization of the common ancestor to multiple living species. It is formulated as the search for a genome that minimizes a rearrangement distance measure among given genomes. Several attempts have been reported for solving this problem. These range from simple heuristic methods to a stochastic local search algorithm inspired by WalkSAT, a well-known local search algorithm for the satisfiability problem in propositional logic. The main objective of our research is to develop improved algorithmic techniques for tackling the genomic median problem. In particular, we have developed an algorithm that is based on tabu search and iterated local search. To alleviate the dependence of the algorithm performance on a single fixed parameter setting, we have included a reactive scheme that automatically adapts the tabu list length of the tabu search part and the perturbation strength of the iterated local search part. In fact, computational results show that our final algorithm reaches very high performance for the genomic median problem and we have found a new best solution for a real-world case.