There is no need to worry—adding cold water to top up your simmering broth will not ruin the flavor or make it cloudy. In fact, what your flatmate mentioned is actually a misunderstanding of stock-making science.
In classical culinary training, adding a splash of cold water or ice to a hot stock pot is actually a recognized technique known as shocking or dérochage. When you introduce cold water to a hot pot, the rapid temperature change helps coagulate dissolved proteins and causes fat and scum to rise directly to the surface, making it much easier to skim off and resulting in a clearer broth, not a cloudier one.
What truly causes broth or bouillon to become cloudy is vigorous, rolling boiling. When liquid boils rapidly, the mechanical agitation emulsifies the rendered fat and microscopic protein particles into the water, trapping them in suspension. As long as you bring the heat back down to a very gentle simmer with just a few lazy bubbles breaking the surface, your extraction and clarity will remain intact.
Topping up with cold tap water simply replaces the water that evaporated too quickly. Cold tap water is actually preferable to hot tap water, as hot water from the heater can sometimes carry dissolved minerals or metallic notes from your plumbing.
For future batches, keep the heat low and steady, skim any foam that rises to the top, and feel completely free to top up with cold water whenever your liquid level gets too low.
