Does a double pole in a mixed correlator imply troubles for the QFT?

Question

It is known that diagonal correlation functions (say propagators) can at most have single poles in their spectrum. I am wondering if the existence of double poles in mixed correlators in a QFT (say for example the correlators between the stress energy tensor and a scalar operator in a conformal field theory $\langle T^{\mu \nu}(q) \phi(q') \rangle$ has any bad implication for the theory.

Comments

There is nothing that would forbid it, I think.

---

There may be many poles in the propagator. Probably, the simplest example is Chern-Simons theory coupled to matter. See formulas (26) and (28), for example, in these lecture notes: http://cds.cern.ch/record/379553/files/9902115.pdf.