Physics-informed neural networks (PINNs), a scientific machine learning framework for solving inverse and forward problems involving partial differential equations, are seeing many successes in fluid research recently. Based on PINNs, this paper develops a physics-informed framework for reconstructing incompressible Reynolds-averaged Navier-Stokes (RANS) solutions from boundary data through the incorporation of Reynolds stress transport (RST) formulations without introducing empirical models for the unclosed terms. Specifically, the RST equations are first formulated with the exact forms for the advection and production terms as derived from the Reynolds averaging process while the unclosed terms that need empirical modeling in traditional turbulence models are absorbed into a single inadequacy term. The resulting underdetermined systems are then embedded via the PINNs approach to reconstruct turbulent flow fields based on the mean velocity components and Reynolds stress components at the domain boundary. A mechanism of enforcing the realizability condition for the Reynolds stress is further proposed to enhance the reconstruction performance, via a customized neural network design. The proposed approach is applied and evaluated via a set of flow cases, including a zero-pressure-gradient turbulent boundary layer flow, an adverse-pressure-gradient turbulent boundary layer flow, and a set of flow cases over periodic hills of parameterized geometries. The results show that for all the test flow cases, the mean flow quantities are reconstructed very accurately, and a decent accuracy is obtained for the turbulence quantities including both the normal and shear Reynolds stress components. The codes for this work will be publicly available in GitHub at https://github.com/zhangxcii/RSTnets.