What happens if you mix two insoluble liquids but with the same density?
There is no "or" in the first place. An emulsion is a two-phased system, it is just that one phase comes in a form of small droplets. Small as they are, however, they are pretty big from molecular point of view.

Now to the point. If you stir the liquids vigorously, you'll have an emulsion. Short of that, they will attempt to minimize the surface, and the shape with minimal surface is a sphere. Indeed, that's what you will end up with. It's been done before. Look up the Plateau's oil drop experiment: they put oil in a water/ethanol mixture with ratio tuned so as to match the density of oil, and behold, it just floats there in a huge spherical drop.

If both liquids get in contact with the wall, the geometry might become more complicated. The phase separation surface will develop certain contact angle where it touches the wall, and the rest of it will form some kind of minimal surface, which can be surprisingly diverse.

So it goes.