Why did my chicken soup go cloudy?

My guess would be that you boiled the soup at some point, possibly for an extended period of time.

If you bring it to a full boil the fat from the meat will emulsify and distribute itself through the liquid. This is the same stuff that foams to the top, the "scum" that a lot of recipes (usually ones that say bring to a boil, then simmer) tell you to skim off. I would venture to say that this distribution of the fat into the liquid is also why it tasted heavier.

When making stock or soups, the most I'll heat them is to just below boiling and keep them at that simmer for a little longer to make up for not bringing it to a boil. When making stock, I strain once through a chinois and again through a piece of cheesecloth in a chinois to help reduce the floating particulates, but this won't really solve the cloudiness problem (though it will help some).

You can also try putting it into the fridge overnight so that the fat comes to the top and solidifies, but I don't think this will solve the problem, only help reduce it.

If it's just the stock that's gone cloudy, this page has some suggestions on how to clarify it, but honestly unless you're presenting it to guests, I see no reason to even bother. I've made cloudy stock and clear stock before and except for a slight "heaviness" difference, the taste is generally about the same. The cloudy stock sometimes has a more oily mouth feel, but it's not a major difference - and I've found some people seem to prefer the soups or rices I make with a cloudy stock.

If you have a hard time controlling the temperature on your range-top, you can use an oven-safe stock pot and cook it in the oven at 180°F to keep it just below a boil.

One other trick, depending on what kind of soup you were making - if you were making (for example) chicken soup, consider trying to turn it into a cream of chicken soup to hide the cloudy broth :)