Taking the directional limits will reveal Mathematica's conventions:
Assuming[x > 1, Limit[EllipticK[x + I ε], ε -> 0, Direction -> 1]]
EllipticK[x]
Assuming[x > 1, Limit[EllipticK[x + I ε], ε -> 0, Direction -> -1]]
2 I EllipticK[1 - x] + EllipticK[x]
