Pseudo-$C_\ell$s for spin-$s$ fields with component-wise weighting

We present a generalisation of the standard pseudo-$C_\ell$ approach for power spectrum estimation to the case of spin-$s$ fields weighted by a general positive-definite weight matrix that couples the different spin components of the field (e.g. $Q$ and $U$ maps in CMB polarisation analyses, or $\gamma_1$ and $\gamma_2$ shear components in weak lensing). Relevant use cases are, for example, data with significantly anisotropic noise properties, or situations in which different masks must be applied to the different field components. The weight matrix map is separated into a spin-0 part, which corresponds to the "mask" in the standard pseudo-$C_\ell$ approach, and a spin-$2s$ part sourced solely by the anisotropic elements of the matrix, leading to additional coupling between angular scales and $E/B$ modes. The general expressions for the mode-coupling coefficients involving the power spectra of these anisotropic weight components are derived and validated. The generalised algorithm is as computationally efficient as the standard approach. We implement the method in the public code NaMaster.