  In this article we consider rough differential equations (RDEs) driven by
non-geometric rough paths, using the concept of branched rough paths introduced
in Gubinelli (2004). We first show that branched rough paths can equivalently
be defined as $\gamma$-H\"older 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 $\mathbf{X}$ lying above a path $X$, there exists a geometric rough path
$\bar{\mathbf{X}}$ lying above an extended path $\bar X$, such that
$\bar{\mathbf{X}}$ contains all the information of $\mathbf{X}$. As a corollary
of this result, we show that every RDE driven by a non-geometric rough path
$\mathbf{X}$ can be rewritten as an extended RDE driven by a geometric rough
path $\bar{\mathbf{X}}$. One could think of this as a generalisation of the
It\^o-Stratonovich correction formula.
