  Tame geometry originated in mathematical logic and implements strong
finiteness properties by defining the notion of tame sets and functions. In
part I we argued that observables in a wide class of quantum field theories are
tame functions and that the tameness of a theory relies on its UV definition.
The aims of this work are (1) to formalize the connection between quantum field
theories and logical structures, and (2) to investigate the tameness of
conformal field theories. To address the first aim, we start from a set of
quantum field theories and explain how they define a logical structure that is
subsequently extended to a second structure by adding physical observables.
Tameness, or o-minimality, of the two structures is then a well-defined
property, and sharp statements can be made by identifying these with known
examples in mathematics. For the second aim we quantify our expectations on the
tameness of the set of conformal field theories and effective theories that can
be coupled to quantum gravity. We formulate tameness conjectures about
conformal field theory observables and propose universal constraints that
render spaces of conformal field theories to be tame sets. We test these
conjectures in several examples and highlight first implications.
