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 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 \gevcc. This analysis is the first to extend the LHC searches for electroweak supersymmetric production of gauginos to high tanβ and slepton next-to-lightest supersymmetric particle scenarios.