++ and -- are mutators - see perldoc overload. This is how to implement them:

package Number::Formatted;
use Number::Format;
my $format = new Number::Format (
    thousands_sep => '.',
    decimal_point => ',' );
use overload
    '""' => sub { $format->format_number( ${$_[0]} ) },
    '+' =>  sub {
        my $num = ref( $_[1] ) ? ${$_[1]} : $_[1];
        $num = ${$_[0]} + $num;
        $num = - $num if $_[2];
        return bless \$num, Number::Formatted;
    },
    '-' =>  sub {
        my $num = ref( $_[1] ) ? ${$_[1]} : $_[1];
        $num = ${$_[0]} - $num;
        $num = - $num if $_[2];
        return bless \$num, Number::Formatted;
    },
    '++' =>  sub { ++ ${$_[0]}; shift },
    '--' =>  sub { -- ${$_[0]}; shift },
;
sub new {
    my $num = $format->unformat_number( $_[1] );
    return bless \$num, $_[0];
}

package main;

my $n = new Number::Formatted ( '1.234,56' );
my $m = new Number::Formatted ( '123,45' );

print $n, "\n";
print $n + 10, "\n";
print $n + $m, "\n";
print $n - $m, "\n";
print ++$n, "\n";

# 1.234,56
# 1.244,56
# 1.358,01
# 1.111,11
# 1.235,56
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