This post is aimed to stimulate some discussions.
We are familiar with many physical descriptions and theories of the (many-body quantum) system, with both quasi-particle description and Lagrangian description. For example:
the Landau Fermi-liquids theory.
Here quasi-particle description are simply a way to find effective excitations for the (many-body quantum) system. The effective excitations may not be the original elementary constituents or elementary particles/spins of the system. These effective excitations contain quasi-particle, quasi-string, quasi-brane excitations, etc.
$\bullet$ My question is that:
what are systems with No quasi-particle description but with Yes Lagrangian description.
what are systems with Yes quasi-particle description but with No Lagrangian description.
what are systems with No quasi-particle description and with No Lagrangian description.
NOTE: For instance, I suppose that,
1+1-dimensional Luttinger liquids are examples of 1. No quasi-particle description but with Yes Lagrangian description.
See, e.g. T. Giamarchi, "Quantum Physics in One Dimension" Chap 2.
Fig. 2.4. The occupation factor $n(k)$ of 1+1D Luttinger liquids. Instead of the usual discontinuity at $k_F$ for a Fermi liquid, it has a power law essential singularity. This is the signaturethat fermionic quasiparticles do not exist in one dimension. Note that the position of the singularity is still at k_F. This is a consequence of Luttinger's theorem.
On the other hand, it is likely that
examples of 2 or examples of 3 happens in RR fields or D-branes of string theory, which has Yes/No quasi-particle description, with NO Lagrangian description.
Eg. see this Ref:Stability of Fermi Surfaces and K-Theory by Horava, see page 1 right column: This implies that the RR ﬁelds are also objects in K theory, and not diﬀerential forms , making the low energy description of string theory on manifold Y in terms of Lagrangian (1) questionable. ... once the p-form $C_p$ are reinterpreted as K-theory objects, it is not clear how to even deﬁne Lagrangian (1). This crisis of the Lagrangian formulation of low-energy string theory is further supported by the discovery  of apparently non-Lagrangian phases in the partition functions of various string and M-theory vacua. Perhaps this means that the Lagrangian framework currently available is insuﬃcient for RR ﬁelds but its suitable generalization awaits to be discovered. (Important steps in this direction have been taken .) Alternatively, the theory may require a non-Lagrangian formulation. (This may already be suggested by the presence of a self-dual RR ﬁeld strength in Type IIB theory). Before we settle on either of these two alternatives, however, we should consider a third possibility. The subtle K-theory features of string theory could be an emergent phenomenon, with D-branes and RR ﬁelds emerging as composites of some more elementary degrees of freedom that admit a conventional Lagrangian description.
What else physical systems and theories are examples of 1. 2. 3.?