| Summary: | A branched rough path X consists of a rough integral calculus for X : [0, T] → 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 valued in a different manifold. These results extend previous work on 3 > p-rough paths [ABCF22], itself a generalisation of the Itô calculus on manifolds developed by Schwartz, Meyer and Émery [Sch82, Mey81, É89, É90], to the setting of non-geometric rough calculus of arbitrarily low regularity
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