Kilopi,
Simple - hydrostatics. The water to bulge out must come from somewhere! It moves from the "sides" of the Earth, relative to the Moon's direction, into the "bulges", so the water level around the sides goes down.

Eroica,
Thank you! I had not realised that the Earth-Moon CoM was where it is, whereupon my argument falls in ruins. Or does it?

I think you have forgotten that as the Earth orbits an internal, eccentric CoM, the orbital speed of points on the Earth's surface relative to the CoM will vary. A point furthest from the Moon, far from the CoM, will travel much faster than one facing the the Moon, near to the CoM. For the same reason, at that sub-Moon point, a particle slightly further out from the CoM, on the surface of the Ocean for instance, will still travel faster than one deeper, so it will be going faster than it needs to stay in 'orbit'. It WILL tend to move away away from the CoM, forming a bulge towards the Moon.

Please shoot me down again, if I deserve it! The "different-orbit" and "gravity-vector" theories (see the Bad Astronomer's explanation) are compatible I believe - they both depend on the same mechanics. If they don't how is that non-rotating satellite tethers work?
John