Additive manufacturing, also known as three-dimensional (3D) printing, offers an innovative approach to fabricating objects layer by layer, providing advantages for rapid prototyping and the production of customized components. Modelling fracture in these components is of great importance to prevent failure in real-world applications and identify optimal configurations through parametrized simulations. Second-order phase-field fracture models have gained popularity due to their capability to capture complex crack patterns and their suitability for implementation using the finite element method. High-order phase-field models increase the regularity of the exact solution and thus improve the convergence behaviour of numerical schemes. However, they require specialized numerical solution techniques and the connection between their high-order terms and the mechanical behaviour of materials is not yet fully understood. In this talk, we introduce a fourth-order phase-field model of fracture in anisotropic materials and present a mixed finite element method for solving the derived fourth-order equation. Specifically, our focus is on interpreting the physical meaning of fourth-order terms within the context of continuum damage theories for anisotropic materials with microstructure. Given that the macroscopic mechanical behaviour of 3D printed samples depends significantly on microstructure defects (caused by the distribution of individual material depositions), the proposed fourth-order phase-field model has the capability to capture changes in fracture patterns, such as crack kinks. Numerical simulations and experiments are presented to assess the performance of the proposed model on 3D printed samples.