Applying a magnetic field to many ferrofluids leads to a significant increase in viscosity, but the phenomenon has yet to find technological exploitation because of the thinning caused by even weak shear flows. We have discovered that the addition of plant-virus-derived nanotubes to a commercial ferrofluid can give rise to a dramatic enhancement in magnetoviscosity and a suppression of shear thinning. The dependence of this effect on nanotube aspect ratio and surface charge, both of which were varied biotechnologically, is consistent with a "scaffolding" of magnetic particles into quasi-linear arrays. Direct support for this explanation is derived from transmission electron micrographs, which reveal a marked tendency for the magnetic nanoparticles to decorate the outside surface of the virus nanotubes.