Consequence of stress-energy tensor vanishing on-shell?

Let’s say we have a QFT, which off-shell has a non-zero stress-energy tensor, but vanishes when the equations of motion are applied.

If the stress-energy tensor vanishes off-shell, then all field operators are necessarily spacetime independent. But what can we say about the fields spacetime dependence when it only vanishes on-shell?

My guess is that the fields can be considered constant at separated points in correlation functions, but receives delta function contributions. So the two-point correlator would be

$$langlePhi_1(x)Phi_2(y)rangle=A_{12}+B_{12}delta(x-y).$$

Where $A_{12}$ is some constant and $B_{12}$ could potentially be some polynomial in derivatives.