Oils and water often spontaneously separate to form two coexisting liquids, one mainly oil, and one mainly water. For example, if you add olive oil to vinegar the two liquids separate out into droplets of oil in the vinegar. But at least for some oils, they mix in at least some proportions with water at higher temperatures, so you can have a single hot mixed liquid, that on cooling separates out into oil droplets in water. This is a well studied and common phenomenon. But what if you simultaneously cool, and mix in more water? For example, what if you start with with hot oil in water, with for example, 20% oil, that is sufficiently hot that water will dissolved all the oil? Then you cool to a lower temperature, where water can only dissolve say 15% oil, but at the same time you mix in an equal volume of pure water?
Cooling is faster than mixing, as heat moves faster than molecules, so the first thing that happens is that in the water with 20% oil, oil droplets form, because at the new colder temperature, water can only dissolve 15% water. So oil droplets form, but as they do so, the pure water mixes into the oil and water mixture. Ultimately, as you mixed equal volumes of 20% oil and water, and pure water, the final oil concentration is only 10%, so all the oil will dissolve.
So the oil droplets are only temporary, you will only briefly have two different liquids. The YouTube above is of a simple computer simulation model of this effect. In the left-hand side, you see green droplets forming, coalescing and growing but only for a short time. As the movie continues to run you will see mixing from the right-hand side dilutes the green droplets away, and they dissolve.
This brief separation does not seem to have been studied very much, but I think there is some interesting physics going on. The are two processes going on, separation of the two liquids, and mixing, and competition between the two will set the size of the droplets that form. The slower the mixing, the larger the droplets can grow before they are dissolved. And at least some of the time, people trying to make protein crystals, observe what may be brief separation, so it may have important consequences.