This dataset contains data supporting the results presented in the paper "Birefringent Anti-resonant Hollow-core Fiber" and the supplementary material. It includes the data used to plot each figure (in .xlsx format), together with simulated data obtained using COMSOL. Hollow-core fibres have demonstrated record performance in applications such as high-power pulse delivery, quantum computing, and sensing. However, their routine use is yet to become reality. A major obstacle is the ability to maintain the polarisation state of light over a broad range of wavelengths, while also ensuring single-mode guidance and attenuation that is low enough for practical applications that require only a few meters of fibre length (<1 dB/m). Here we simulated, fabricated and characterized a single-mode birefringent anti-resonant hollow-core fibre. The birefringence was achieved by introducing capillary tubes of different thicknesses, thereby creating reduced symmetry in the structure. The measured group birefringence is in good agreement with the calculated group birefringence from simulations across the fibre guidance band within the telecommunications C-band. At 1550 nm, we measured a group birefringence of 4.4E-5, which corresponds to a phase birefringence of 2.5E-5. The measured loss of the fibre was 0.46 dB/m at 1550 nm. The measured polarisation extinction ratio of the fibre at 1550 nm was 23.1 dB (25.7 dB) along the x-(y-) polarisation axis, relating to an h-parameter of 9.8E-4 (5.3E-4).