good => [ 'ID3' ],
hello => [ 'ID1', 'ID3', 'ID2' ],
goodbye => [ 'ID1' ],
where => [ 'ID1' ]


##</code><code>##

use strict;
use warnings;

my $h =  {
  ID1 => {
    myStr => 'hello',
    description => 'Description 1',
    yourStr => [ 'goodbye', 'where', 'hello']
  },
  ID2 => {
    myStr => 'good',
    description => 'Description 2',
    yourStr => [ 'hello', ]
  },
  ID3 => {
    myStr => 'testvar',
    description => 'Description 2',
    yourStr => [ 'hello', 'good' ]
  },				  
};

my %yourstr;
for my $id ( keys %$h ) {
  for ( @{ $h->{$id}->{yourStr} } ) {
    push @{ $yourstr{$_} }, $id;
  }
}

for my $id ( keys %$h ) {
  my $mystr = $h->{$id}->{myStr};
  if ( $yourstr{$mystr} ) {
    my @matches = grep { $_ ne $id } @{ $yourstr{$mystr} };
    print "'$mystr' in $id matches @matches\n";
  }
}


##</code><code>##

'hello' in ID1 matches ID3 ID2
'good' in 1D2 matches ID3