Tour of the Merrimack series by Rebecca M. Meluch -
(Quotes from a Tor.com review by SF author Jo Walton)
There are five of them The Myriad (2003), Wolf Star (2005), The Sagittarius Command (2007), Strength and Honor (2009) and The Ninth Circle (2011).
There are a number of things about them that are absolutely wonderful. There are also other things about them that are just cringe-inducingly terrible. I can’t think of anything else offhand that needs me to forgive so much but which remains worth it. [...]
To start with, there are Romans in space. And there are American marines in space. And in a logical universe they’d be fighting each other, but in the universe they find themselves in they are allied against amazingly horrific FTL tentacled aliens. Now, if that doesn’t already have you sitting up straighter in your chair, these books are probably not for you. But there’s also an awesome narrative element in play in these books, which is what makes me really like them.
You know why there are Romans in space, in our future? It’s because the Roman Empire never died, it just became a huge underground conspiracy, running the world in secret until there was a suitable terraformed planet, Palatine, whereupon they grabbed it and sent out the call and lots of doctors and lawyers and classics majors up and left Earth to head for Nova Roma to re-found the empire. (The only reason I know that this conspiracy doesn’t really exist is that if it did, I would be in it. See you on Palatine!)
(Note that since this review was written, but before this answer, a 6th book The Twice and Future Caesar (2015) was added.)