I am probably going to get slammed for this, because it violates everything we were taught about tidal forces, but the antipodal tide is caused by the centrifugal force created by the Earth's rotation about the earth/moon barycenter, not differential gravitational forces. While the moon's gravity is less on the side of the earth furthest from it, that force is still towards the moon, not away from it.
There is a good explanation of this centrifugal force explanation on NOAA's website here. I'll cite some pictures and pertinent text, but this site is a good read for anyone trying to understand the forces responsible for the Earth's tides.
Here is a diagram showing the earth and moon's movement around the system's barycenter:
And some of the pertinent text:
The center of revolution of this motion of the earth and moon around their common center-of-mass lies at a point approximately 1,068 miles beneath the earth's surface, on the side toward the moon, and along a line connecting the individual centers-of-mass of the earth and moon. (see G, Fig. 1) The center-of-mass of the earth describes an orbit (E1, E2, E3..) around the center-of-mass of the earth-moon system (G) just as the center-of-mass of the moon describes its own monthly orbit (M1, M2, M3..) around this same point.
- The Effect of Centrifugal Force. It is this little known aspect of the moon's orbital motion which is responsible for one of the two force components creating the tides. As the earth and moon whirl around this common center-of-mass, the centrifugal force produced is always directed away from the center of revolution. All points in or on the surface of the earth acting as a coherent body acquire this component of centrifugal force. And, since the center-of-mass of the earth is always on the opposite side of this common center of revolution from the position of the moon, the centrifugal force produced at any point in or on the earth will always be directed away from the moon. This fact is indicated by the common direction of the arrows (representing the centrifugal force Fc) at points A, C, and B in Fig. 1, and the thin arrows at these same points in Fig. 2.
And finally another diagram that the blockquote cites:
Yes, one can probably find lots and lots of quotations declaring that the antipodal tide is caused because the moon's gravitational force is much less on the far side than on the near side, but that doesn't make it true. And I believe NOAA ought to be a pretty authoritative source. If they can't get it right...