Consideration of the way Perl converts a string to a number is interesting, but I was puzzled by why all the other things that are done to $N in TEST() seem to have no effect. Here's the narrative I've imagined for the critical part of what's happening:

1. In the  $N += 0; statement, the number 0 is added to the string $N (after its conversion to a temporary number according to the rules already discussed in this thread) to generate a number that is assigned to a temporary, intermediate variable, but not yet to $N.
2. In the course of numifying the string and adding, the "... isn't numeric in addition ..." warning is triggered, and this is trapped by the  local $SIG{__WARN__} = sub { ... } handler temporarily installed by TEST().
3. The  $SIG{__WARN__} handler assigns a lot of stuff to $N and prints a lot of stuff, but none of it makes any difference because...
4. When the  $N += 0; statement completes, it assigns the contents of its temporary, intermediate variable to $N and that's all, folks; any changes to $N prior to this final assignment are wiped out.

c:\@Work\Perl\monks>perl
use strict;
use warnings;

sub TEST {

  defined $_[0] or return 0;
  my $N = shift;
  printf "TEST('$N') called - ";

  local $SIG{__WARN__} = sub {
    # doing anything or nothing to $N has no effect here
  # return;  # try this
    printf "(in warn '$N' -> ";
    $N =~ tr|0-9||cd;
    $N = "0$N";
    $N = 'garbage';
    printf "'$N') - ";
    };

  printf "\$N is '$N' before += - ";
  $N += 0;
  printf "\$N is '$N' after += \n";
  return int($N);

  }

for my $n (qw(55 55x x55)) {
  my $m = TEST($n);
  print "TEST('$n') returns $m \n\n";
  }

__END__
TEST('55') called - $N is '55' before += - $N is '55' after +=
TEST('55') returns 55

TEST('55x') called - $N is '55x' before += - (in warn '55x' -> 'garbag
+e') - $N is '55' after +=
TEST('55x') returns 55

TEST('x55') called - $N is 'x55' before += - (in warn 'x55' -> 'garbag
+e') - $N is '0' after +=
TEST('x55') returns 0
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Update: Re-writing the  $N += 0; statement as  $N = $N + 0; may allow better visualization of what I think happens: The  $N + 0 expression must be evaluated first, and it's in this evaluation that the warning is triggered and a bunch of irrelevant things are done to $N; then the  $N = ...; assignment is done, overwriting any changes to $N made by the  $SIG{__WARN__} handler.