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Using file_get_content('ht*p://offsite.com/login/'); works fine, except that all the internal links then point to the containing page, not the target.
The target links should read ht*p://offsite.com/linked_page/ but instead they all construe to ht*p://mysite.com/linked_page/ and results in an "Object not found!" error.
If you want to replace them you would have to add the site in before the links.
you could try replacing
$show_shop = file_get_contents ('ht*p://astore.amazon.com/login-20/');
$show_shop = str_replace("\"/login-20/","\"ht*p://astore.amazon.com/login-20/", $show_shop);
echo $show_shop ;
Now the shop appears at it's own URL and page, and the links are all rewritten from relative to absolute.
The only problem now is that aStore reverts to the Amazon site whenever a link is clicked - but that's a whole new challenge!