In Perl, you also have to handle the condition of a runtime error which is not an object

I believe that the try block is where the memory leak issues usually show up, since that is where the bulk of your working code is. But why throw catch away too?

sub catch (&) {
    if ($@) {
        my ($catch) = @_;    
        my $e = $@;
        $e = My::Base::Exception->new($@) 
            unless (ref($@) && UNIVERSAL::isa($@, "My::Base::Exception
+"));
        $catch->($e);
    }
}

eval {
    die "test";
};
catch {
    my $e = shift;
    if ($e->isa('IOException')) {
        # ...
    }
    else {
        print $e;
    }
};
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