Well it's not clear if you want multiple matches to be listed separately, like "lacy smith" twice.

If not I'd just construct an "or"ed regex with all search terms.

 
   DB<127> @search=qw/smith sanders/
 => ("smith", "sanders")

  DB<128> $regex= join "|", @search
 => "smith|sanders"

  DB<129> @matches = grep { /($regex)/ } @full
 => (
  "Bob-smith#anthropology",
  "Sandy-smith#sociology",
  "Tom-sanders#english",
  "Anthony-sanders#english",
)

  DB<130> @full
 => (
  "Bob-smith#anthropology",
  "Sandy-smith#sociology",
  "Tom-sanders#english",
  "Tina-lacy#socialwork",
  "Anthony-sanders#english",
)
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update

improved code

BTW we appreciate if you post real code not just multiply broken pseudo stuff. :)