  The presence of nearby conformal field theories (CFTs) hidden in the complex
plane of the tuning parameter was recently proposed as an elegant explanation
for the ubiquity of "weakly first-order" transitions in condensed matter and
high-energy systems. In this work, we perform an exact microscopic study of
such a complex CFT (CCFT) in the two-dimensional $O(n)$ loop model. The
well-known absence of symmetry-breaking of the $O(n>2)$ model is understood as
arising from the displacement of the non-trivial fixed points into the complex
temperature plane. Thanks to a numerical finite-size study of the transfer
matrix, we confirm the presence of a CCFT in the complex plane and extract the
real and imaginary parts of the central charge and scaling dimensions. By
comparing those with the analytic continuation of predictions from Coulomb gas
techniques, we determine the range of validity of the analytic continuation to
extend up to $n_g \approx 12.34$, beyond which the CCFT gives way to a gapped
state. Finally, we propose a beta function which reproduces the main features
of the phase diagram and which suggests an interpretation of the CCFT as a
liquid-gas critical point at the end of a first-order transition line.
