We study the axial (magnetic) tidal Love numbers of a Schwarzschild black hole surrounded by a spherically symmetric matter distribution. While the formalism developed here is general, we specialize to the case of anisotropic fluids as a proxy for dark matter distributions, computing the Love numbers for different density profiles of astrophysical interest. We employ two complementary methods: a small-compactness expansion, yielding closed-form analytic expressions, and direct numerical integration of the perturbation equations. We discuss the connection between different formulations of the fluid perturbations and the resulting Love numbers. We further show that density profiles lacking compact support generically produce logarithmic terms in the asymptotic expansion of the perturbation variable, which obstruct the standard tidal matching procedure and whose origin we trace to the absence of a strictly vacuum exterior. Our findings highlight the importance of controlling the asymptotic structure of the matter distribution when defining tidal observables for black holes dressed by matter, and provide a general framework that can be applied to other spherically symmetric environments.

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• Dark matter
• Gravitational waves