A. Novelli, F. Piacentini, S. Micheli, E. Allys, A. Anand, J. Aumont, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, A. Basyrov, A. Besnard, D. Blinov, M. Bortolami, T. Brinckmann, F. Cacciotti, E. Calabrese, P. Campeti, A. Carones, F. Carralot, F. J. Casas, J. Chandran, K. Cheung, M. Citran, L. Clermont, F. Columbro, A. Coppolecchia, F. Cuttaia, P. de Bernardis, T. de Haan, E. de la Hoz, M. De Lucia, P. Diego-Palazuelos, H. K. Eriksen, F. Finelli, C. Franceschet, U. Fuskeland, G. Galloni, M. Galloway, M. Gerbino, M. Gervasi, R. T. Génova-Santos, T. Ghigna, S. Giardiello, C. Gimeno-Amo, A. Gruppuso, M. Hazumi, S. Henrot-Versillé, L. T. Hergt, E. Hivon, H. Ishino, K. Kohri, L. Lamagna, M. Lattanzi, C. Leloup, F. Levrier, A. I. Lonappan, M. López-Caniego, G. Luzzi, J. Macias-Perez, V. Maranchery, E. Martínez-González, S. Masi, S. Matarrese, T. Matsumura, M. Migliaccio, M. Monelli, G. Morgante, L. Mousset, R. Nagata, T. Namikawa, P. Natoli, F. Noviello, L. Pagano, A. Paiella, D. Paoletti, G. Pascual-Cisneros, G. Patanchon, G. Piccirilli, M. Pinchera, G. Polenta, L. Porcelli, M. Remazeilles, A. Ritacco, M. Ruiz-Granda, Y. Sakurai, L. Salvati, J. Sanghavi, V. Sauvage, D. Scott, M. Shiraishi, G. Signorelli, R. M. Sullivan, Y. Takase, L. Terenzi, M. Tomasi, M. Tristram, L. Vacher, B. van Tent, P. Vielva, S. Vinzl, I. K. Wehus, G. Weymann-Despres, E. J. Wollack, M. Zannoni

One of the current challenges in observational cosmology is obtaining high-precision polarisation maps of the CMB to measure primordial $B$-modes and constrain the tensor-to-scalar ratio ($r$). The weakness of this signal compared to foregrounds and $E$-to-$B$ leakage makes this task particularly challenging, requiring large detector arrays operating at multiple frequencies and extremely precise calibration. We present a cross-calibration algorithm to determine relative calibration of detectors within the same frequency band of a CMB experiment. The method iteratively compares single-detector maps with band-averaged maps and can be applied to any calibration parameter that can be observed and corrected at the map level, relaxing pre-flight calibration requirements and enabling post-processing validation. We validate the pipeline by calibrating the polarisation angle of simulated LiteBIRD observations, including both random detector miscalibration and wafer-level rotations. The algorithm converges to correct values with arcminute precision. Finally, we propagate residual calibration uncertainties through component separation and tensor-to-scalar ratio estimation pipelines using both parametric (FgBuster) and blind (HILC) methods. The induced bias on $r$ remains well below the LiteBIRD systematics budget of $δr < 6.5\times10^{-6}$, demonstrating that the method is suitable for next-generation CMB experiments.