We investigate the cosmological dynamics of multiple tachyon fields with inverse square potentials. A phase-space analysis of the spatially flat FRW models shows that there exist power-law cosmological scaling solutions. We study the stability of the solutions only for the case of two tachyon fields and find that the potential kinetic-scaling solution is a global attractor. However, in the presence of a barotropic fluid the solution is an attractor only in one region of the parameter space, and the tracking solution is an attractor in the other region. We briefly discuss the physical consequences of these results.