First introduced in the 60s, positivity bounds for S-matrices have undergone a revival in the past few years, with systematic implementations that improve by the day. Among the recent results are sharp bounds on the EFTs of various gauge and gravitational theories. A common feature of the resulting exclusion regions, however, is that they tend to be saturated by S-matrices that are physically not that interesting. In this talk, I will discuss what assumptions can take us beyond this paradigm. After a brief recap on the subject, I will describe how positivity bounds define a bootstrap problem closely analogous to the well-known conformal bootstrap. This analogy will motivate new assumptions, which we will then apply to large N confining gauge theories and to weakly-coupled completions of 10d supergravity. I will explain how this refinement leads to peculiar solutions which are, respectively, surprisingly close to real-world QCD and tree-level string theory.