I just started reading about BV formalism and I have a question regarding the slogan/philosophy of taking the derived critical locus of a smooth function. Specifically, as explained here, the BV complex to a Lagrangian field theory (LFT) serves the purpouse of calculating the derived critical locus $C_{dS=0}$ of the action functional $S$ and then the derived/homotopy quotient by the gauge group.
My question is: How is this justified? Is there an example of classical LFTs where one has to consider the derived critical locus instead of the usual one?