  We reformulate the theory of p-adic iterated integrals on semistable curves
using the unipotent log rigid fundamental group. This fundamental group carries
Frobenius and monodromy operators whose basic properties are established. By
identifying the Frobenius-invariant subgroup of the fundamental group with the
fundamental group of the dual graph, we characterize Berkovich--Coleman
integration, which is path-dependent, as integration along the
Frobenius-invariant lift of a path in the dual graph. We demonstrate the basic
properties for this integration theory. Vologodsky's path-independent
integration theory which was previously described using a monodromy condition
can now be identified as Berkovich--Coleman integration along a combinatorial
canonical path arising from the theory of combinatorial iterated integration as
developed by the first-named author and Cheng. We use this definition to lay
the groundwork for computing iterated integrals.
