use strict;
use warnings;

my $string = 'This is "the search" string "that was" supplied';

my @quotes;

while( $string =~ s["(.*?)"][] )
{
    push( @quotes, $1 );
}

my @keys = split( ' ', $string );

use Data::Dumper;
print "Quotes:\n", Dumper( \@quotes );
print "Keys:\n", Dumper( \@keys );

output:

Quotes:
$VAR1 = [
          'the search',
          'that was'
        ];
Keys:
$VAR1 = [
          'This',
          'is',
          'string',
          'supplied'
        ];