Conducting composite materials, with electrical percolation threshold as low as 0.05–0.08 wt %, are prepared by incorporating multiwall carbon nanotube (CNT) in melt‐mixed co‐continuous polystyrene (PS)/ethylene‐vinyl acetate (EVA) copolymer (70:30 wt/wt) blend. CNT is found to be preferentially localized within the EVA phase, characterizing a double percolation phenomenon, which accounts for outstanding electrical conductivity (around 0.5 S/m with the addition of 1 wt % of CNT). The noncovalent functionalization of CNT with alkylphosphonium‐based ionic liquid results in composites with improved electrical conductivity (2.5 S/m with 1 wt % of CNT) and better electromagnetic interference shielding effectiveness. Both nonfunctionalized CNT (n‐CNT) or CNT with alkylphosphonium‐based ionic liquid contribute for a refinement of the co‐continuous structure, as indicated by scanning electron microscopy. The dispersion of CNTs and their physical networked structure within the polymer matrix are suggested by transmission electron microscopy and melt rheological experiments. © 2017 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2018, 135, 45564.