We present a combined neutron diffraction (ND) and high-field muon spin rotation ($mu$SR) study of the magnetic and superconducting phases of the high-temperature superconductor La$_{1.94}$Sr$_{0.06}$CuO$_{4+y}$ ($T_{c} = 38$~K). We observe a linear dependence of the ND signal from the modulated antiferromagnetic order (m-AFM) on the applied field. The magnetic volume fraction measured with $mu$SR increases linearly from 0% to $sim$40% with applied magnetic field up to 8~T. This allows us to conclude, in contrast to earlier field-dependent neutron diffraction studies, that the long-range m-AFM regions are induced by an applied field, and that their ordered magnetic moment remains constant.
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