The exchange is an artifact of the formalism. When we write it down, we tend to label the particles as 1 and 2 (or something else), but physically there is no different between the particles. They are not "labeled" in any way in the physical world. Therefore, we need rules for our formalism when we exchange the particles with the different labels. Note, the signs do not come from a phase. It is produced by the statistics, i.e., commutation of anti-commutation relations.

The exchange is an artifact of the formalism. When we write it down, we tend to label the particles as 1 and 2 (or something else), but physically there is no difference between the particles. They are not "labeled" in any way in the physical world. Therefore, we need rules for our formalism when we exchange the particles with the different labels. Note, the signs do not come from a phase. It is produced by the statistics, i.e., commutation or anti-commutation relations.