  Precipitating a notion emerging from recent research, we formalise the study
of a special class of compact quantum metric spaces. Abstractly, the additional
requirement we impose on the underlying order unit spaces is the Riesz
interpolation property. In practice, this means that a `quantum metric Choquet
simplex' arises as a unital $\mathrm{C}^*$-algebra $A$ whose trace space is
equipped with a metric inducing the $w^*$-topology, such that tracially
Lipschitz elements are dense in $A$. This added structure is designed for
measuring distances in and around the category of stably finite classifiable
$\mathrm{C}^*$-algebras, and in particular for witnessing metric and
statistical properties of the space of (approximate unitary equivalence classes
of) unital embeddings of $A$ into a stably finite classifiable
$\mathrm{C}^*$-algebra $B$. Our reference frame for this measurement is a
certain compact `nucleus' of $A$ provided by its quantum metric structure. As
for the richness of the metric space of isometric isomorphism classes of
classifiable $\mathrm{C}^*$-algebraic quantum metric Choquet simplices
(equipped with Rieffel's quantum Gromov--Hausdorff distance), we show how to
construct examples starting from Bauer simplices associated to compact metric
spaces. We also explain how to build non-Bauer examples by forming `quantum
crossed products' associated to dynamical systems on the tracial boundary.
Further, we observe that continuous fields of quantum spaces are obtained by
continuously varying either the dynamics or the metric. In the case of deformed
isometric actions, we show that equivariant Gromov--Hausdorff continuity
implies fibrewise continuity of the quantum structures. As an example, we
present a field of deformed quantum rotation algebras whose fibres are
continuous with respect to a quasimetric called the quantum intertwining gap.
