Text this: Is the fermionic exchange phase also acquired locally?

  _____     _____     _  _      ___     ______   
 / ____||  |  ___||  | \| ||   / _ \\  |      \\ 
/ //---`'  | ||__    |  ' ||  / //\ \\ |  --  // 
\ \\___    | ||__    | .  || |  ___  |||  --  \\ 
 \_____||  |_____||  |_|\_|| |_||  |_|||______// 
  `----`   `-----`   `-` -`  `-`   `-` `------`