Nginx – Chrome ignores Set-Cookie directive from nginx reverse proxy

I want to expose v1.2 of my API (and it's documentation) to users in China. The parent site mysite.mycompany.com is blocked for them, but 1.2.3.4 isn't, so I'm building a reverse proxy at that address that they can use to access my service.

I've configured an Nginx to reverse proxy http://mysite.mycompany.com to http://1.2.3.4

The proxy works great in Firefox and IE, but not in Chrome. Login is managed using cookies, but Chrome ignores the Set-Cookie header sent by the proxy during the login, and I don't understand why. This prevents Chrome users from logging in.

The parent site's response headers to the login request ( where Set-Cookie succeeds and login is successful) are

HTTP/1.1 302 Found
Cache-Control: no-cache, must-revalidate, post-check=0, pre-check=0
Content-Type: text/html; charset=UTF-8
Date: Tue, 12 Aug 2014 18:16:57 GMT
ETag: "1407867417"
Expires: Sun, 19 Nov 1978 05:00:00 GMT
Last-Modified: Tue, 12 Aug 2014 18:16:57 +0000
Location: http://mysite.mycompany.com/user/me/apps
Server: Apache
X-Powered-By: PHP/5.3.15
Content-Length: 0
Connection: Keep-Alive
Set-Cookie: SESS25fe1d0b3b7239f403e22223da0614cf=Oo-PXr4XYQCFUKYUyZgX3UJpWADAhF_q7Xf-8gko3BU; expires=Thu, 04-Sep-2014 21:50:17 GMT; path=/; domain=.mysite.mycompany.com; HttpOnly

while the proxy site (where Set-Cookie fails and login is not successful) has the following response headers:

HTTP/1.1 302 Found
Server: nginx/1.4.7
Date: Tue, 12 Aug 2014 18:57:42 GMT
Content-Type: text/html; charset=UTF-8
Content-Length: 0
Location: http://1.2.3.4/user/me/apps
Cache-Control: no-cache, must-revalidate, post-check=0, pre-check=0
ETag: "1407869862"
Expires: Sun, 19 Nov 1978 05:00:00 GMT
Last-Modified: Tue, 12 Aug 2014 18:57:42 +0000
X-Powered-By: PHP/5.3.15
Connection: Keep-Alive
Set-Cookie: SESS25fe1d0b3b7239f403e22223da0614cf=DrY_4Y1NUPdIv9MsXPMjCr7Bzkq7DjyOmQPcNhetKXU; expires=Thu, 04-Sep-2014 22:31:02 GMT; path=/; domain=.1.2.3.4; HttpOnly

The interesting part of my nginx config is

http {
    # Ngnix defaults up to here then

    proxy_cache_path /var/spool/nginx keys_zone=CACHE:20m levels=1:2 inactive=6h max_size=1g;

    server {
        listen  80;


        location /v1.2/ {
                proxy_pass  http://api.mysite.mycompany.com/v1.2/;
                proxy_set_header  X-Real-IP    $remote_addr;
                proxy_set_header  X-Real-HOST  $host;
        }


        location / {
                proxy_pass      http://mysite.mycompany.com;
                # Required to prevent the host serving gzipped pages that can't be modified with sub_filter
                proxy_set_header        Accept-Encoding ''; 
                proxy_set_header        X-Real-IP $remote_addr;
                proxy_set_header        X-Forwarded-For $proxy_add_x_forwarded_for;
                proxy_set_header        X-Real-HOST  $host;
                proxy_cookie_domain     mysite.mycompany.com     1.2.3.4;
                sub_filter      "mysite.mycompany.com"   "1.2.3.4";
                sub_filter_once off;
                proxy_cache     CACHE;
        }

    }

}

I've tried with the caching elements turned off, but it doesn't seem to change anything.

Do you have any idea why Chrome might be refusing to set the proxy's cookie?