  We construct states describing Bose Einstein condensates at finite
temperature for a relativistic massive complex scalar field with
$|\varphi|^4$-interaction. We start with the linearised theory over a classical
condensate and construct interacting fields by perturbation theory. Using the
concept of thermal masses, equilibrium states at finite temperature can be
constructed by the methods developed in arXiv:1306.6519 and arXiv:1502.02705.
Here, the principle of perturbative agreement plays a crucial role. The
apparent conflict with Goldstone's Theorem is resolved by the fact that the
linearised theory breaks the $U(1)$ symmetry, hence the theorem applies only to
the full series but not to the truncations at finite order which therefore can
be free of infrared divergences.
