Nothing feels a reaction force to the Coriolis force because Newton's laws only apply in an inertial frame. If we imagine a frame with some sort of wild, random acceleration, then in that frame at a given moment it would look like, in a room with completely stationary tables and chairs and things, everything was simultaneously accelerating the same direction, and there would be huge "fictitious forces" needed to explain this. There are no third-law pairs going on, though. The fictitious forces were necessary to make $F=ma$ work out in this accelerating frame, but by introducing them we sacrificed the third law (at least in regards to those kinematic forces).
Imagine dropping a ball off a tower. It gets deflected east.
If we analyze this scenario from a frame rotating along with the Earth, there is no conflict with Newton's third law because it does not hold for the Coriolis force or other kinematic forces.
If we analyze the scenario from an inertial frame in which the Earth is rotating, the trajectory simply is not deflected. The ball lands east of where it was dropped because its velocity in a tangential direction, which comes from the rotation of the Earth, stayed the same, but as it fell its angular velocity $v/r$ increased because $r$ decreased. That led it to "deflect east" when it was really moving in a perfectly ordinary parabola.