The electrical conductivity of a macroscopic assembly of nanomaterials is determined through a complex interplay of electronic transport within and between constituent nano-objects. Phonons play dual roles in this situation: their increased populations tend to reduce the conductivity via electron scattering, while they can boost the conductivity by assisting electrons to propagate through the potential-energy landscape. We identified a phonon-assisted coherent electron transport process between neighboring nanotubes in temperature-dependent conductivity measurements on a macroscopic film of armchair single-wall carbon nanotubes. Through atomistic modeling of electronic states and calculations of both electronic and phonon-assisted junction conductances, we conclude that phonon-assisted conductance is the dominant mechanism for observed high-temperature transport in armchair carbon nanotubes. The unambiguous manifestation of coherent intertube dynamics proves a single-chirality armchair nanotube film to be a unique macroscopic solid-state ensemble of nano-objects promising for the development of room-temperature coherent electronic devices.