Trying to implement it according to your remarks leads me to something I don't like at all.

EDIT: See comments in uglier_foo.

my $missing = sub {...};

# private sub
sub ugly_foo ($self, $foo=$missing->(), $bar=$missing->()) {
    say "foo is missing" if $missing->($foo);
    say "bar is missing" if $missing->($bar);
}

# compare with eq
sub uglier_foo($self, $foo=MISSING, $bar=MISSING) {
    # This comes from posting insufficiently tested code at midnight
    # say "foo is missing" if !$foo || $foo eq MISSING;
    # say "bar is missing" if !$foo || $foo eq MISSING;

    say "foo is missing" if $foo && $foo eq MISSING;
    say "bar is missing" if $bar && $bar eq MISSING;
}

# slurping
sub ugliest_foo ($self, @args) {
    my ($foo, $foo_missing);
    if (@args) {
         $foo = shift @args;
    } else {
        $foo_missing = 1;
    }
    my ($bar, $bar_missing);
    if (@args) {
        $bar = shift @args;
    } else {
        $bar_missing = 1;
    }
    if (@args) {
        croak "too many arguments";
    }
    say "foo is missing" if $foo_missing;
    say "bar is missing" if $bar_missing;
}
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So, yes, I stubbornly stick to my solution.