  New Z' gauge bosons arise in many extensions of the Standard Model and
predict resonances in the dilepton invariant mass spectrum. Searches for such
resonances therefore provide important constraints on many models of new
physics, but the resulting bounds are often calculated without interference
effects. In this work we show that the effect of interference is significant
and cannot be neglected whenever the Z' width is large (for example because of
an invisible contribution). To illustrate this point, we implement and validate
the most recent 139 fb$^{-1}$ dilepton search from ATLAS and obtain exclusion
limits on general Z' models as well as on simplified dark matter models with
spin-1 mediators. We find that interference can substantially strengthen the
bound on the Z' couplings and push exclusion limits for dark matter simplified
models to higher values of the Z' mass. Together with this study we release the
open-source code ZPEED, which provides fast likelihoods and exclusion bounds
for general Z' models.
