We introduce a novel method for numerical spin glass investigations: Simulations of two replica at fixed temperature, weighted to achieve a broad distribution of the Parisi overlap parameter
q (multioverlap). We demonstrate the feasibility of the approach by studying the 3D Edwards-Anderson Ising ( ${J}_{\mathrm{ik}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ifmmode\pm\else\textpm\fi{}1$) spin glass in the broken phase ( $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$). This makes it possible to obtain reliable results about spin glass tunneling barriers. In addition, our results indicate a nontrivial scaling behavior of the canonical
q distributions not only at the freezing point but also deep in the broken phase.