Eggs – What are common techniques for adding eggs to soup?

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 :)

Typically you should use yellow onions for cooking. They have a higher sulphur content and are more flavorful after cooking. Raw, a red onion will taste more pungent. However, once cooked it is more mild and sweet than a soup would require. Red, as well as other sweeter onions, have more sugar and water than their yellow counterparts. For more details on the profile, seriouseats has good primur.

If you wish to improve your results, you might slice more thinly and add some salt in advance of sweating (rather than before adding broth) to draw out the excess moisture, as the onion's moisture (more prevalent with red onions) will inhibit browning, ergo inhibiting caramelization and the maillard reaction that break the bigger sugars into little ones and lend flavor through browning. The explanation at norecipes also adds that deglazing with sherry can slow down the process if you are experiencing uneven browning. Another possible tactic would be adding a 1/4 teaspoon of baking soda per pound chopped onion; this will speed up browning.

Ultimately, unless the recipe calls for red onions, you are going to get better results with yellow onions in terms of raw, "sweet and tangy" onion-y flavor. You can use the tips above to try to overcome that, but a nice big bag of yellow onions are going to do you better in the long run, and if you put the same effort into the it will take less time, taste better and be less expensive.