We present a search for chargino-neutralino associated production using like electric charge dilepton events collected by the CDF II detector at the Fermilab Tevatron in proton-antiproton collisions at root s = 1.96 TeV. One lepton is identified as the hadronic decay of a tau lepton, while the other is an electron or muon. In data corresponding to 6.0 fb(-1) of integrated luminosity, we obtain good agreement with standard model predictions and set limits on the chargino-neutralino production cross section for simplified gravity-and gauge-mediated models. As an example, assuming that the chargino and neutralino decays to taus dominate, in the simplified gauge-mediated model we exclude cross sections greater than 300 fb at 95% credibility level for chargino and neutralino masses of 225 GeV/c(2). This analysis is the first to extend the LHC searches for electroweak supersymmetric production of gauginos to high tan beta and slepton next-to-lightest supersymmetric particle scenarios.