  Soon after its theoretical prediction, striped-density states in the presence
of synthetic spin-orbit coupling were realized in Bose-Einstein condensates of
ultracold neutral atoms [J.-R. Li et al., Nature \textbf{543}, 91 (2017)]. The
achievement opens avenues to explore the interplay of superfluidity and
crystalline order in the search for supersolid features and materials. The
system considered is essentially made of two linearly coupled Bose-Einstein
condensates, that is a pseudo-spin-$1/2$ system, subject to a spin-dependent
gauge field $\sigma_z \hbar k_\ell$. Under these conditions the stripe phase is
achieved when the linear coupling $\hbar\Omega/2$ is small against the gauge
energy $m\Omega/\hbar k_\ell^2<1$ . The resulting density stripes have been
interpreted as a standing-wave, interference pattern with approximate
wavenumber $2k_\ell$. Here, we show that the emergence of the stripe phase is
induced by an array of Josephson vortices living in the junction defined by the
linear coupling. As happens in superconducting junctions subject to external
magnetic fields, a vortex array is the natural response of the superfluid
system to the presence of a gauge field. Also similar to superconductors, the
Josephson currents and their associated vortices can be present as a metastable
state in the absence of gauge field. We provide closed-form solutions to the 1D
mean field equations that account for such vortex arrays. The underlying
Josephson currents coincide with the analytical solutions to the sine-Gordon
equation for the relative phase of superconducting junctions [C. Owen and D.
Scalapino, Phys. Rev. \textbf{164}, 538 (1967)].
