  We present a combined neutron diffraction and high field muon spin rotation
($\mu$SR) study of the magnetically ordered and superconducting phases of the
high-temperature superconductor La$_{1.94}$Sr$_{0.06}$CuO$_{4+y}$ ($T_{\rm c} =
37.5(2)$~K) in a magnetic field applied perpendicular to the CuO$_2$ planes. We
observe a linear field-dependence of the intensity of the neutron diffraction
peak that reflects the modulated antiferromagnetic stripe order. The magnetic
volume fraction extracted from $\mu$SR data likewise increases linearly with
applied magnetic field. The combination of these two observations allows us to
unambiguously conclude that stripe-ordered regions grow in an applied field,
whereas the stripe-ordered magnetic moment itself is field-independent. This
contrasts with earlier suggestions that the field-induced neutron diffraction
intensity in La-based cuprates is due to an increase in the ordered moment. We
discuss a microscopic picture that is capable of reconciling these conflicting
viewpoints.
