anti-Stokes line ($\mathrm{Re}(S_1-S_2)=0$)

With the equal-magnitude convention, an anti-Stokes line has $\mathrm{Re}(S_1-S_2)=0$ for competing exponentials. For $e^{\pm x^2/\epsilon }$, its rays are $\arg x=\pi /4+m\pi /2$. Crossing one interchanges which exponential grows. Some literature interchanges the names, so the action-based convention should always be specified.