In altermagnets, time-reversal symmetry breaking spin polarizes electronic states, while total magnetization remains zero. In addition, at altermagnetic surfaces Rashba–spin orbit coupling is activated due to broken inversion symmetry, introducing a competing spin-momentum locking interaction. Here we show that their interplay leads to the formation of complex, chiral spin textures that offer nonlinear spin-to-charge conversion properties. Whereas altermagnetic order suppresses the canonical linear in-plane Rashba-Edelstein response, we establish the presence of an anomalous transversal Edelstein effect for planar applied electric and magnetic field or, alternatively, an in-plane magnetization. Additionally, we predict a purely electric-field-driven nonlinear out-of-plane magnetization. We compute the anomalous response within a general altermagnet d-wave model, with parameters extracted from the ab initio electronic structure of an altermagnetic bilayer. Our results suggest altermagnetic surfaces as a promising platform for unconventional spintronic functionalities.