  We give a formalism for approximate isomorphism in continuous logic
simultaneously generalizing those of two papers by Ben Yaacov and by Ben
Yaacov, Doucha, Nies, and Tsankov, which are largely incompatible. With this we
explicitly exhibit Scott sentences for the perturbation systems of the former
paper, such as the Banach-Mazur distance and the Lipschitz distance between
metric spaces. Our formalism is simultaneously characterized syntactically by a
mild generalization of perturbation systems and semantically by certain
elementary classes of two-sorted structures that witness approximate
isomorphism. As an application, we show that the theory of any
$\mathbb{R}$-tree or ultrametric space of finite radius is stable, improving a
result of Carlisle and Henson.
