cross domain localstorage with javascript

How about using cross domain postmessage and iframes?

So on your wrong-domain-page you include an iframe that posts messages with the cookie data back.

Here is a solid example of cross domain postmessages:
http://blog.teamtreehouse.com/cross-domain-messaging-with-postmessage

live example:
http://codepen.io/anon/pen/EVBGyz //forked sender code with a tiiiiiny change 🙂 :

window.onload = function() {
    // Get the window displayed in the iframe.
    var receiver = document.getElementById('receiver').contentWindow;

    // Get a reference to the 'Send Message' button.
    var btn = document.getElementById('send');

    // A function to handle sending messages.
    function sendMessage(e) {
        // Prevent any default browser behaviour.
        e.preventDefault();

        // Send a message with the text 'Hello Treehouse!' to the new window.
        receiver.postMessage('cookie data!', 'http://wrong-domain.com');
    }

    // Add an event listener that will execute the sendMessage() function
    // when the send button is clicked.
    btn.addEventListener('click', sendMessage);
}

Receiver code:

window.onload=function(){
    var messageEle=document.getElementById('message');
    function receiveMessage(e){
        if(e.origin!=="http://correct-domain.com")
        return;
        messageEle.innerHTML="Message Received: "+e.data;
    }
    window.addEventListener('message',receiveMessage);
}