Background: This question is inspired by Why is a relativistic quantum theory of a finite number of particles impossible?
1 - QFT is typically used to calculate relativistic scattering. The ground state of relativistic QFT is the vacuum, which is an invariant state under Lorentz transformations. This is not the case when we are in a medium (finite density): a finite-density medium in the ground state is not Lorentz invariant (from the point of view of Lorentz transformations, the "finite-density" ground state is different from the vacuum, see the attempted answer below).
2 - On the other hand, we have non-relativistic many-body Quantum Mechanics that is used in condensed matter for finite-density systems. This can be expressed in a "second quantization" fashion, becoming a non-relativistic QFT. Moreover, we also have a sort of non-relativistic "thermal QFT" that allows us to calculate the thermodynamic equilibrium properties of many-body quantum systems (i.e., many-body non-relativistic quantum mechanics after Wick rotation).
Question: How about a "relativistic QFT for finite-density systems"? Namely, a QFT that aims at describing a dense medium of relativistic particles (not just scattering in vacuum). Do we have such a theory or the only thing we have are some "approximations"? By "approximation" I mean a theoretically problematic situation resembling that of the "early relativistic QM" (i.e. the so-called relativistic quantum mechanics), which was then replaced by relativistic QFT.
Answer: Exercise V.2.3 of Zee's QFT in a Nutshell (2nd edition) asks us to develop "QFT at finite density", which would be exactly the framework I am looking for: we have to add a chemical potential term to the Lagrangian in the path integral formalism. Moreover, Zee comments that "finite density, as well as finite temperature, breaks Lorentz invariance", see also this article. Some extra useful and related questions are Chemical potential in quantum field theories, Quantum field theory with a constraint: energy-momentum conservation?. The relation between the use of Lagrange multipliers to impose a certain density and path integral is discussed here. Interesting posts on Lagrange multipliers in QFT: QFTs which are pure constraint, Can auxiliary fields be thought of as Lagrange multipliers?, see also this.
Collection of references: Thermal/finite temperature quantum field theory: online lectures and best books.