Direct Multisearch (DMS) is a well-established class of methods for multiobjective derivative-free optimization, where constraints are addressed by an extreme barrier approach, only evaluating feasible points. In this work, we propose a filter approach, combined with an inexact feasibility restoration step, to address constraints in the DMS framework. The filter approach treats feasibility as an additional component of the objective function, avoiding the computation of penalty parameters or Lagrange multipliers. The inexact restoration step attempts to generate new feasible points, contributing to prioritize feasibility, a requirement for the good performance of any filter approach. Theoretical results are provided, analyzing the different types of sequences of points generated by the new algorithm, and numerical experiments on a set of nonlinearly constrained biobjective problems are reported, stating the good algorithmic performance of the proposed approach.