Will a clustering method solve this noisy matching problem?

A matching problem came to me, recently.

Suppose that there are four lists of data, and the dimension of each data point is three. Each list is generated from a different method. Ideally, only one data point in each list is equivalent to the other (supposing the red data points). We call it the solution, and others are pseudo-solutions. So, it will be easy to find a real solution. We even do not need the entire data but only two data lists. By comparing every possible combination of two data points from the two lists, the solution will be found easily.

All the data contains noise in the real world, unfortunately. The real solutions will be unequal in different lists. Sometimes, the real solution is lost from its list. Even worse, there exist two pseudo-solutions that are surprisingly close to each other, which makes finding the closest data points is useless.

How can I find a real solution in the real world? I can imagine that if two data points are closest but the other three are not that close. The last three is chosen. The reason can be the number of their neighbors is bigger. However, how close two data points should be considered as neighbors?

Will a clustering method solve this noisy matching problem?