  We study approximate $\aleph_0$-categoricity of theories of beautiful pairs
of randomizations, in the sense of continuous logic.
  This leads us to disprove a conjecture of Ben Yaacov, Berenstein and Henson,
by exhibiting $\aleph_0$-categorical, $\aleph_0$-stable metric theories $Q$ for
which the corresponding theory $Q_P$ of beautiful pairs is not approximately
$\aleph_0$-categorical, i.e., has separable models that are not isomorphic even
up to small perturbations of the smaller model of the pair. The theory $Q$ of
randomized infinite vector spaces over a finite field is such an example.
  On the positive side, we show that the theory of beautiful pairs of
randomized infinite sets is approximately $\aleph_0$-categorical. We also prove
that a related stronger property, which holds in that case, is stable under
various natural constructions, and formulate our guesswork for the general
case.
