In all QFTs I know, the Lagrangian density is completely invariant under the Poincare group, $$ \mathcal L \to \mathcal L. $$ On the other hand, the action would be invariant even if the Lagrangian density were invariant only up to a derivative of something, e.g. $$ \mathcal L \to \mathcal L + \partial_1 K. $$ (In this case the action is preserved since $\int d^4x \,\,\partial_1 K(x)=0$.)
Are there any important or interesting QFTs for which the extra term in the Lagrangian density is non-zero, but the action is invariant under the Poincare group?
Remark: There is a somewhat analogous situation in QFT's with super-symmetry: the super-symmetry transformations preserve the action, but do change the Lagrangian density.
