  Let $Y$ be a (partial) minimal model of a scheme $V$ with a cluster
structure. Under natural assumptions, for every choice of seed we associate a
Newton--Okounkov body to every divisor on $Y$ supported on $Y \setminus V$ and
show that these Newton--Okounkov bodies are positive sets in the sense of
Gross, Hacking, Keel and Kontsevich \cite{GHKK}. This construction essentially
reverses the procedure in loc. cit. that generalizes the polytope construction
of a toric variety to the framework of cluster varieties.
  In a closely related setting, we consider cases where $Y$ is a projective
variety whose universal torsor $\text{UT}_Y$ is a partial minimal model of a
scheme with a cluster structure of type $\mathcal A$. If the theta functions
parametrized by the integral points of the associated superpotential cone form
a basis of the ring of algebraic functions on $\text{UT}_Y$ and the action of
the torus $T_{\text{Pic}(Y)^*}$ on $\text{UT}_Y$ is compatible with the cluster
structure, then for every choice of seed we associate a Newton--Okounkov body
to every line bundle on $Y$. We prove that any such Newton--Okounkov body is a
positive set and that $Y$ is a minimal model of a quotient of a cluster
$\mathcal A$-variety by the action of a torus.
  Our constructions lead to the notion of the intrinsic Newton--Okounkov body
associated to a boundary divisor in a partial minimal model of a scheme with a
cluster structure. This provides a wide class of examples of Newton-Okoukov
bodies exhibiting a wall-crossing phenomenon in the sense of Escobar--Harada
\cite{EH20}.
  This approach includes the partial flag varieties that arise as minimal
models of cluster varieties. For the case of Grassmannians, our approach
recovers, up to interesting unimodular equivalences, the Newton--Okounkov
bodies constructed by Rietsch--Williams in \cite{RW}.
