Over Christmas, as is traditional, I have been eating and drinking a lot. The drinks include beer, wine, dessert wine, liqueurs, etc, but no ouzo, pastis or sambuca. Ouzo, pastis and sambuca all have the distinctive flavour of aniseeed. While Googling something for a paper I am drafting I came across the explanation for why these drinks become cloudy when you add water. Ouzo for example is quite strong, about 40%, and so it is often diluted with water, but as soon as the water is added, the ouzo goes from clear to cloudy.
It is cloudy or milky for the same reason as milk, they both scatter light, i.e., light pinballs around inside the ouzo-plus-water mixture. Some of this gets absorbed and some bounces back into our eyes. The light is scattered, i.e., rays of light have their direction changed, by tiny droplets in the diluted ouzo or milk. In milk these are fat droplets. In ouzo they form mainly from one of the molecules that gives the very distinctive aniseed flavour of ouzo; this is trans-anethole.
Trans-anethole is an oil in the sense that it does not mix or dissolve in water. But it does dissolve in alcohol (ethanol), and so it dissolves in sufficiently strong alcohol/water mixtures. So at 40% alcohol the trans-anethole is soluble and so the ouzo, or pastis etc, is clear but when water is added, and the alcohol is diluted, there is not enough alcohol to dissolve the trans-anethole and it comes out of solution as small droplets. These scatter light and the drink turns cloudy.
Why the droplets are so small appears to be unknown. If you shake, for example, balsamic vinegar (which is mainly water) and olive oil together the olive oil droplets rapidly grow and you quickly get large, visible to the naked eye, oil droplets. But diluted ouzo is different, the oil droplets remain tiny. The Ouzo, pastis, etc are complex mixtures, and it may be that growth of the droplets is greatly slowed by something else in the drink.