  A set of conditions that any effective field theory needs to satisfy in order
to allow for the existence of a viable UV completion has recently gained
attention in the cosmological context under the name of $\textit{positivity
bounds}$. In this paper we revisit the derivation of such bounds for Horndeski
gravity and translate them into a complete set of viability conditions in the
language of effective field theory of dark energy. We implement the latter into
$\texttt{EFTCAMB}$ and explore the large scale structure phenomenology of
Horndeski gravity under positivity bounds. We build a statistically significant
sample of viable Horndeski models, and derive the corresponding predictions for
the background evolution, in terms of $w_{\rm DE}$, and the dynamics of linear
perturbations, in terms of the phenomenological functions $\mu$ and $\Sigma$,
associated to clustering and weak lensing, respectively. We find that the
addition of positivity bounds to the traditional no-ghost and no-gradient
conditions considerably tightens the theoretical constraints on all these
functions. The most significant feature is a strengthening of the correlation
$\mu\simeq\Sigma$, and a related tight constraint on the luminal speed of
gravitational waves $c^2_T\simeq1$. In anticipation of a more complete
formulation of positivity conditions in cosmology, this work demonstrates the
strong potential of such bounds in shaping the viable parameter space of
scalar-tensor theories.
