Moose type constraints are not strings, they are instances of Moose::Meta::TypeConstraint.

Sure, I assumed they have semantics beyond 'being a String'. I just meant from the syntactic perspective, Moose::Manual::Types pretty much makes it look like the types are strings. At least

...
isa => Str | ArrayRef,
...
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doesn't work without MooseX::Types (Bareword "Str" not allowed while "strict subs"...)

Anyway, I tried to make this inherited union constraints "do what I mean" for multiple inheritance and came up with

package ValAtom;

use strict;
use warnings;
use 5.10.1;

use Moose;
use MooseX::Types::Moose -all;

extends 'KeyAtom', 'JustTesting';

has '+data' => (
    isa     => __PACKAGE__->_existing_constraint('data') . ' | ArrayRe
+f | HashRef',
);

sub _existing_constraint {
        my ($class, $attr) = @_;
        return join ' | ', map {
            $_->meta->find_attribute_by_name($attr)->type_constraint
        } $class->meta->superclasses;
    }

1;
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Back to string again - obviously - as I couldn't figure out how to do it otherwise. But this seems to work.