  A branched rough path $X$ consists of a rough integral calculus for $X \colon
[0, T] \to \mathbb R^d$ which may fail to satisfy integration by parts. Using
Kelly's bracket extension [Kel12], we define a notion of pushforward of
branched rough paths through smooth maps, which leads naturally to a definition
of branched rough path on a smooth manifold. Once a covariant derivative is
fixed, we are able to give a canonical, coordinate-free definition of integral
against such rough paths. After characterising quasi-geometric rough paths in
terms of their bracket extension, we use the same framework to define
manifold-valued rough differential equations (RDEs) driven by quasi-geometric
rough paths. These results extend previous work on $3 > p$-rough paths
[ABCRF22], itself a generalisation of the Ito calculus on manifolds developed
by Meyer and Emery [Mey81, E89, E90], to the setting of non-geometric rough
calculus of arbitrarily low regularity.
