I have heard this one several times but was never given a convincing explanation.
Is this a real phenomenon? (Bonus points for a good physical description of the behavior or reasons it couldn't occur.)
(If you are interested in reading further about the details of the Coriolis Effect itself, I would encourage you to read some of the Wikipedia article, as it goes into great detail.)
As several sources indicate, the Coriolis Effect is not strong enough in bodies of water such as those in bathtubs, sinks, toilets, etc. to reliably influence their rotational direction as they drain. The rotational direction of water in such containers is much more likely to be affected by any combination of the following:
The Wikipedia article on the Coriolis Effect does mention a 1908 experiment in which a physicist successfully demonstrated the effect with a large tub of water (more than 1000 liters). When the plug was pulled, the water eventually began to rotate in a counter-clockwise direction (same as hurricanes in the northern hemisphere, where the experiment took place). However, the tub used was circular, with a tiny central drain, and the experimenter took great care to eliminate as many disturbances as possible from the water before draining it.
So, if your sink/toilet/bathtub is sufficiently large, circular, has a tiny central drain, and you let the water completely stop moving before you drain it, you would observe that the rotational direction of the water as it drains is influenced by the Coriolis Effect (i.e. varies by hemisphere). But for the other 99.9999% of the sinks/toilets/bathtubs in the world, the direction in which the water rotates is determined by something else.
The Coriolis Effect is dependent on velocity, mass and latitude.
Water in a sink is indeed subjected to the Coriolis Effect (just like every object moving along the surface of the earth) but the "force" is way too small to influence how it swirls down the drain.
To quickly demonstrate that you just have to throw a basket ball, walk in a straight line or steer a super tanker. The ball, you and the giant ship all have a mass considerably larger than the water in the sink, yet there is no need to course correct for Coriolis. Other 'forces' (e.g. wind) are usually more of a nuisance than Coriolis.
So even though Coriolis does have an effect on the water in the sink, it can easily be demonstrated that it's just too small to be responsible for how it swirls. Otherwise we would all have a hard time not walking in circles.
The coriolis effect is correct, but the explanation fails to mention that the circulation you get is reversed if the fluid is being pushed outward from the center, rather than being pulled inward. That's why in the northern hemisphere, atmospheric lows have counter-clockwise circulation, and highs have clockwise, and it's reversed in the southern hemisphere.
I think another useful explanation is angular momentum. There is the usual image of a figure skater, when pulling in her arms and legs, spins faster, and slows down when extending them. Similarly, fluids at "rest" are actually sharing in the angular rotation of the Earth, which is counterclockwise in the northern hemisphere (to someone looking down toward the surface), and clockwise in the southern. So if the fluid is pulled into a center, that angular velocity is increased, and decreased if pushed outward. Of course, if some other force (like manually stirring the bucket) gives the fluid a different angular velocity, that will override the earth's angular velocity.
If you're wondering what happens near the equator, check this out.
Edit: An afterthought explanation: In rotational terms, the surface of the earth at the equator, and the air with it, is traveling eastward at about 1000 mph (25,000 m / 24 hr). At 60 degrees latitude, north or south, it is traveling half that speed, and at the poles, zero. So a parcel of air moving away from the equator is going to want to keep going eastward at the speed it was going, and if it's toward the equator it's going want to go west.
Since a low pressure area pulls air into it's center, air coming from lower latitude will tend to move east, while air coming from higher latitude will tend to move west. There's your circulation.
Water doesn't swirl at all unless it's moved, so the answer to your actual question is no.
However, if you correct for all other factors, water being drained from a large container rotates about the exit in a certain direction based on its location on earth.
Watch both simultaneously.