When an elided constituent and its antecedent do not match syntactically, the presence of a word implying the non-actuality of the state of affairs described in the antecedent seems to improve the example. ("This information should be released but Gorbachev didn't." vs. "This information was released but Gorbachev didn't.") We model this effect in terms of Non-Actuality Implicatures (NAIs) conveyed by non-epistemic modals like "should" and other words such as "want to" and "be eager to" that imply non-actuality. We report three studies. A rating and interpretation study showed that such implicatures are drawn and that they improve the acceptability of mismatch ellipsis examples. An interpretation study showed that adding a NAI trigger to ambiguous examples increases the likelihood of choosing an antecedent from the NAI clause. An eye movement study shows that a NAI trigger also speeds on-line reading of the ellipsis clause. By introducing alternatives (the desired state of affairs vs. the actual state of affairs), the NAI trigger introduces a potential Question Under Discussion (QUD). Processing an ellipsis clause is easier, the processor is more confident of its analysis, when the ellipsis clause comments on the QUD. (Contains 3 tables.)