What exactly is the driving force of a vapor turning into a liquid once it tips just above the saturated state?

The volume on the x axis is a weighted average of the liquid and vapor specific volumes, weighted in terms of the fraction of liquid water and water vapor present at the particular quality of the mixture. The quality is defined as the fraction of the total mass present that is vapor.

My main question is once it enters this zone where the temperature and pressure are set at a saturated value, what is the driving force for the conversion of the vapor to liquid.

Some type of "seed", as in crystalization.

In crystallization Nucleation is the step where the solute molecules or atoms dispersed in the solvent start to gather into clusters, on the microscopic scale (elevating solute concentration in a small region), that become stable under the current operating conditions. These stable clusters constitute the nuclei. Therefore, the clusters need to reach a critical size in order to become stable nuclei .

So in this case what happens is that energy is given up by the molecules binding into clusters that are the start of crystallization.

The corresponding mechanism from gas to liquid again needs a nucleation process, for water for example

Nucleation is typically defined to be the process that determines how long an observer has to wait before the new phase or self-organized structure appears. Nucleation is often found to be very sensitive to impurities in the system. Because of this, it is often important to distinguish between heterogeneous nucleation and homogeneous nucleation. Heterogeneous nucleation occurs at nucleation sites on surfaces in the system. Homogeneous nucleation occurs away from a surface.

Again, energy will be radiated away when molecules start to bind to the liquid form( that is why it gets warmer when it rains).

Cloud chambers works this way, by having supersaturated vapor and nucleation supplied by the ions generated by charge particles.

Repeatability then depends on how clean surfaces are , or how many nucleation site impurieties there are.

My main question is once it enters this zone where the temperature and pressure are set at a saturated value, what is the driving force for the conversion of the vapor to liquid?

A wide range of liquid and vapor volumes can exist at a given equilibrium temperature and pressure based on the Antoine equation, from a container that is practically full of liquid with a small vapor bubble in it, to a container that is practically full vapor with a small liquid droplet in it. Given this fact, the energy balance of this system must still be maintained.

It is standard practice in chemical engineering to select a standard temperature and pressure (e.g., ideal gas at 0 K), and assign a "heat content" of zero to this standard state. Substances that have a higher temperature and pressure than the standard state are said to have a heat content, or enthalpy (and yes, I'm well aware that physicists do not define heat this way). For the example of water, this concept leads to the fact that a given mass of water has less heat content that the same mass of steam, and that heat content obviously differs by the heat of vaporization. This means that for a given container with a mixture of water and steam, the mass of water multiplied by its heat content plus the mass of steam multiplied by its heat content must equal the total heat content of the system. Any heat added to the system shifts the equilibrium toward more steam and less water, and any heat removed from the system shifts the equilibrium toward more water and less steam. Thus, the driving force for the proportion of liquid vs. vapor is the total heat content of the system, and whether or not heat is being transferred into or out of the system.