A communication network can be modeled as a directed connected graph with edge weights that characterize performance metrics such as loss and delay. Network tomography aims to infer these edge weights from their pathwise versions measured on a set of intersecting paths between a subset of boundary vertices, and even the underlying graph when this is not known. Recent work has established conditions under which the underlying directed graph can be recovered exactly the pairwise Path Correlation Data, namely, the set of weights of intersection of each pair of directed paths to and from each endpoint. Algorithmically, this enables us to consistently fused tree-based view of the set of network paths to and from each endpoint to reconstruct the underlying network. However, in practice the PCD is not consistently determined by path measurements. Statistical fluctuations give rise to inconsistent inferred weight of edges from measurement based on different endpoints, as do operational constraints on synchronization, and deviations from the underlying packet transmission model. Furthermore, ad hoc solutions to eliminate noise, such as pruning small weight inferred links, are hard to apply in a consistent manner that preserves known end-to-end metric values. This paper takes a unified approach to the problem of inconsistent weight estimation. We formulate two type of inconsistency: \textsl{intrinsic}, when the weight set is internally inconsistent, and \textsl{extrinsic}, when they are inconsistent with a set of known end-to-end path metrics. In both cases we map inconsistent weight to consistent PCD within a least-squares framework. We evaluate the performance of this mapping in composition with tree-based inference algorithms.