Republican socialism?
Yes, it is entirely possible. Spain recently tried it, or started to try it, when Zapatero the socialist adopted republicanism in the style of Princeton philosopher Phillip Pettit. Zapatero fell from grace after the 2008 crisis, but mainly because *the left* rejected his market-friendly response. His republicanism apparently wasn’t socialistic enough. The fascinating story is told in this book.
On Pettit-style republicanism, see this previous post.