[quote=@Valiance]
<Snipped quote by Legend>

In this lab, we had a 300 mL beaker of tap water. We first make it basic by adding some bleach to it. After this, we pour a bismuth solution into the water-bleach combo, and a white cloudiness appears. The question is: What makes the white cloudiness? I already know it has somwthing to do wth the formation if BiOCl, but I'm not sure about anything beyond that.
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BiClO forms a precipitate (or simply a separate solution. If it has solid particles, it's a precipitate) that clouds the water solution. I don't believe Bi reacts with Hydrogen Hydroxide, so it will only do a single replacement reaction with the bleach, and contrast against the clarity of Hydrogen Hydroxide.