 Ghost propagator for a negative derivative kinetic term

With signature $(+---)$ and action $-\int {\partial }^{\mu }\bar{c}{\partial }_{\mu }c$, integration by parts gives the quadratic kernel $K_c=\square$, or $-p^2$ in Fourier space. The Grassmann Gaussian integral with odd sources gives $Z=\exp [-i\bar{\eta }{K}_{c,F}^{-1}\eta ]$. Left and right source derivatives, including their insertion factors, yield $⟨Tc\bar{c}⟩=i{K}_{c,F}^{-1}$ and hence the displayed sign. Replacing $\bar{c}$ by $-\bar{c}$ changes this propagator and the ghost-gluon vertex together; it is a convention change, not a physical discrepancy.