I set up some test pages to see how this behaved, and it looks like you do need to install SSL on domain-b.com. Most browsers I tested blocked the nonsecure content. Others gave a warning. The same things happened if domain-b.com had a self-signed certificate. The connection was still blocked.
The simplest and cheapest solution I found was to install a free SSL certificate from StartSSL on domain-b.com. This CA seems to be recognized by the browsers I tested on so the connection to domain-b.com works fine. If anyone sees any issues with this that I'm not thinking of, please let me know.