 Gamma-five convention without an imaginary unit

For the mostly-minus Minkowski metric, this product anticommutes with every gamma matrix and is anti-Hermitian in the standard representation. The conventional Hermitian chirality matrix is $iG$. Nevertheless $\bar{\psi }G\psi$ is Hermitian, because ${\gamma }^{0}G{\gamma }^0=-G$. Therefore a real interaction coefficient in $-g\varphi \bar{\psi }G\psi$ is compatible with Hermitian dynamics and gives the Feynman vertex $-igG$.