What governs the merging of globs of inmiscible liquid? If I add a small amount of chloroform to a bunch of water, it falls to the bottom as a glob.

If I shake the container vigorously (being carefully not to spill any of the dangerous substance!), the globs of chloroform break apart into many tiny globules. These globules immediately sink to the bottom and collect together by the action of gravity the moment I stop agitating my mixture.

If I watch this pile of globules for some time, I observe that they tend to merge after several seconds, perhaps even a minute or more for some. I suspect that if I reached in with a needle and prodded the globules, they would merge faster. Also, I suspect that if I were to tabulate how long each individual globule survives before merging, I would obtain a half-life curve.

What governs the merging of these globules (I'm guessing surface tension)? Why don't they instantly merge? Why do a few little globules sometimes persist for many minutes, stuck to the side of a much larger globule?
Emulsion chemistry is actually rather complicated, but I'll try to give a conceptual description. Whenever you have a surface dividing two phases, there is an energy associated with creating a certain amount of surface, expressed as a surface energy density with dimensions of energy divided by area. This represents the energy cost of disrupting the intermolecular interactions of each phase at the interface, e.g. for your water/chloroform system, water can't hydrogen bond across the interface so it takes energy to break all the bonds required to make the surface. The larger the surface area, the more energy it takes. Logically, you can see that the lowest energy state is the one in which the interfacial area is minimized, which is why in the absence of external forces like gravity, liquids assume spherical shapes which give the smallest surface area for a given volume.

When you have many small droplets, there is a large surface area to volume ratio which is a high energy state, so it is energetically favourable for the droplets to form larger droplets. However, the reason it doesn't occur instantaneously is that an energy barrier can exist between the two states, for various reasons (electrostatic repulsion is a big one). In most cases, smaller droplets are lost first because the energy barrier to coalescing is surface area dependant. Larger droplets are more stable. Depending on the liquids, different separation mechanisms can occur with different kinetics. (coalescence, Ostwald ripening) I can't give a definite answer, but I'm guessing in your system you would not see an exponential decay-type half-life as that assumes a stochastic process where the probability is always the same. Since the stability depends on surface area, this isn't true in this case.