In the high-Tc cuprate La2−xSrxCuO4 (LSCO), sound velocity and attenuation measurements have recently been shown to be sensitive to the antiferromagnetic (AFM) glass phase , a glass-like spin freezing with antiferromagnetic correlations. Here we focus on the case of La1.88Sr0.12CuO4  where this magnetic phase is the most prominent . Our ultrasound data reveal classical signatures of spin freezing that are reinforced up to unexpectedly high magnetic fields. This indicates very strong competition between superconductivity and magnetic freezing, tuned by the magnetic field. Comparison between different acoustic modes points toward a bilinear coupling of B1g strain with the spin fluctuations, which suggest that they have a nematic character.