The conductivity in multiphase polymer systems containing conductive nanoinclusions is presumed to be dominated by phase continuity and nanoparticle percolation phenomena. In this work, we show that the interfacial bridging of multiwalled carbon nanotubes (MWCNTs) results in an additional two-decade increase in the conductivity for a multiphase polymer system that was already highly continuous and where the MWCNT concentration was above the percolation threshold. The localization of MWCNTs in polylactic acid/poly(butylene adipate-co-terephthalate) (PLA/PBAT) blends is studied at two different PBAT compositions of 20 and 50 vol %. Surface energy analysis indicates that the thermodynamic equilibrium localization of MWCNTs should be in the PBAT phase. Transmission electron microscopy images show that when MWCNTs are added to the PLA/PBAT melt or are initially dispersed into the PBAT phase, almost all MWCNTs remain in the PBAT phase. However, when the nanotubes are initially dispersed into the PLA phase and then melt-mixed with PBAT, partitioning of MWCNTs between phases is observed with a significant proportion embedded in PBAT. In addition, a number of MWCNTs are found to form a hair-like structure extending away from the interface into both phases. These dangling MWCNTs at the interface allow for interface bridging between separated PBAT domains or ligaments. The migration and localization of MWCNTs from PLA to PBAT is discussed in detail. It is postulated that the hair-like interface and bridging of MWCNTs are related to both a slow migration velocity across the PLA/PBAT interphase region because of the system's low interfacial tension, as well as to MWCNT rearrangement within the PBAT phase. It is shown that the interface bridging of MWCNTs increases electrical conductivity even in the sample with a double percolated structure. This bridging effect contributes to both the perfecting of incomplete PBAT continuity and the creation of additional conductive pathways. The obtained results indicate the potential of interface bridging as a mechanism to significantly increase the maximum achievable electrical conductivity in low interfacial tension multiphase systems with MWCNTs.