Carbon nanotubes (CNTs) are under intense investigation in materials science owing to their potential for modifying the electrical conductivity sigma, shear viscosity eta, and other transport properties of polymeric materials. These particles are hybrids of filler and nanoscale additives because their lengths are macroscopic whereas their cross-sectional dimensions are closer to molecular scales. The combination of extended shape, rigidity and deformability allows CNTs to be mechanically dispersed in polymer matrices in the form of disordered 'jammed' network structures. Our measurements on representative network-forming multiwall nanotube (MWNT) dispersions in polypropylene indicate that these materials exhibit extraordinary flow-induced property changes. Specifically, sigma and eta both decrease strongly with increasing shear rate, and these nanocomposites exhibit impressively large and negative normal stress differences, a rarely reported phenomenon in soft condensed matter. We illustrate the practical implications of these nonlinear transport properties by showing that MWNTs eliminate die swell in our nanocomposites, an effect crucial for their processing.