The detection of the radio emission following a neutrino interaction in ice is a promising technique to obtain significant sensitivities to neutrinos with energies above PeV. The detectable radio emission stems from particle showers in the ice. So far, detector simulations have considered only the radio emission from the primary interaction of the neutrino. For this study, existing simulation tools have been extended to cover secondary interactions from muons and taus. We find that secondary interactions of both leptons add up to 25% to the effective volume of neutrino detectors. Also, muon and tau neutrinos can create several detectable showers, with the result that double signatures do not constitute an exclusive signature for tau neutrinos. We also find that the background of atmospheric muons from cosmic rays is non-negligible for in-ice arrays and that an air shower veto should be considered helpful for radio detectors.