Real nodal lines (RNLs) featuring real Chern numbers and second-order boundary modes have attracted widespread attention. In all previously reported realizations, an RNL is linked by another nodal line, and whether an RNL can be linked by other types of band degeneracies has remained open. Here, we propose a second-order real nodal-line semimetal in which a pair of RNLs is linked by a nodal surface. We show that this state can be realized in spinless systems with both PT and nonsymmorphic S_2zT symmetries, where the S_2zT-enforced nodal surface prevents the pair of RNLs from annihilation. Each nodal line carries a nontrivial real Chern number ν_R=1, giving rise to topological hinge Fermi arcs located at a pair of PT-related hinges. Guided by this construction, we identify the interpenetrated graphene network (IGN) as a promising material realization. First-principles calculations confirm that a pair of nodal lines traversing the Brillouin zone are linked by a nodal surface and each nodal line carries double nontrivial Z₂ charges. The bulk-boundary correspondence of IGN manifests as a pair of hinge Fermi arcs together with drumhead surface states. Our work establishes nodal surfaces as a new linking partner for real nodal lines and provides a roadmap for exploring higher-order real topology in carbon-based and other light-element systems.