In this article we consider rough differential equations (RDEs) driven by non-geometric rough paths, using the concept of branched rough paths introduced in (J. Differential Equations 248 (2010) 693-721). We first show that branched rough paths can equivalently be defined as gamma-Holder continuous paths in some Lie group, akin to geometric rough paths. We then show that every branched rough path can be encoded in a geometric rough path. More precisely, for every branched rough path X lying above a path X, there exists a geometric rough path (X) over bar lying above an extended path (X) over bar, such that (X) over bar contains all the information of X. As a corollary of this result, we show that every RDE driven by a non-geometric rough path X can be rewritten as an extended RDE driven by a geometric rough path (X) over bar. One could think of this as a generalisation of the Ito-Stratonovich correction formula.