How to copy posts from one site to another in real time (not multisite)

I'm working to rebrand my company's website (new look, URL, etc.). I've already transferred all of the existing posts, pages, users, media from site B (old) to site A (new) and I have site A up and running for me to design around now.

My question is this:
Is there a way that when a user publishes a post on site B that it can also create that post on site A, so when I eventually deactivate site B all of the posts, images, etc. on site A will be current?

I've searched around a bit and I see there are articles on how to perform this within a multisite, but we are using two individual installs. I've also seen RSS methods or plugins, but those seem to be more of a feed, rather than creating actual database entries in the new database.