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One of the most relevant problems in Bioinformaticsand Computational Biology is the search and reconstruction ofthe most accurate phylogenetic tree that explains, as exactly aspossible, the evolutionary relationships among species from agiven dataset. Different criteria have been employed to evaluatethe accuracy of evolutionary hypothesis in order to guide a searchalgorithm towards the best tree. However, these criteria may leadto distinct phylogenies, which are often conflicting among them. Therefore, a multi-objective approach can be useful. In this work, we present a phylogenetic adaptation of a multiobjective variablemesh optimization algorithm for inferring phylogenies, to tacklethe phylogenetic inference problem according to two optimalitycriteria: maximum parsimony and maximum likelihood. Theaim of this approach is to propose a complementary view ofphylogenetics in order to generate a set of trade-off phylogenetictopologies that represent a consensus between both criteria. Experiments on four real nucleotide datasets show that ourproposal can achieve promising results, under both multiobjectiveand biological approaches, with regard to other classical andrecent multiobjective metaheuristics from the state-of-the-art.