The advancement of 3D printed concrete (3DPC) technology, particularly reinforcement integration, has increased the demand for reliable predictive tools to assess structural performance in the hardened state. Unlike cast concrete, 3DPC exhibits pronounced mechanical anisotropy due to its layer-wise deposition process, interlayer interfaces, and process-dependent parameters like printing speed and interlayer time intervals. Accurately capturing this anisotropy and the associated cracking and failure mechanisms remains a major challenge for numerical modeling and structural design. Although several numerical approaches exist, many rely on simplifying assumptions or require explicit resolution of interlayer regions, causing high computational costs and limited structural-scale applicability. Furthermore, multiphysics effects inherent to the additive manufacturing process are still insufficiently addressed. We develop a phase-field-based cohesive zone modeling framework for 3DPC that provides an efficient and physically grounded description of anisotropic fracture behavior. The approach captures the directional dependence of cracking and failure induced by the additive manufacturing process through a structurally motivated, direction-dependent tensor formulation. This enables a smeared macroscale representation of anisotropy without explicitly resolving individual interlayer interfaces, ensuring computational efficiency while retaining essential physical features. A key aspect of the model is its ability to account for the influence of interlayer printing time, which governs the mechanical properties of interlayer regions. By incorporating this process-related parameter into the constitutive description, the framework directly links manufacturing conditions to macroscopic fracture behavior. The model is calibrated and validated against experimental tensile and shear tests on 3DPC specimens printed with different interlayer time intervals. Numerical results show good agreement with measured mechanical responses, capturing both strength anisotropy and stiffness degradation. Moreover, the framework reproduces experimentally observed crack initiation and propagation patterns, demonstrating its ability to represent failure mechanisms relevant to structural applications. Overall, the proposed approach provides a robust and efficient tool for modeling anisotropic fracture in 3DPC and supports the design and optimization of additively manufactured concrete structures.