  A spin-orbit coupled two-dimensional (2D) Bose gas is shown to simultaneously
possess quasi and true long-range order in the total and relative phase
sectors, respectively. The total phase undergoes a Berenzinskii-
Kosterlitz-Thouless transition to a low temperature phase with quasi long-range
order, as expected for a two- dimensional quantum gas. Additionally, the
relative phase undergoes an Ising-type transition building up true long-range
order, which is induced by the anisotropic spin-orbit coupling. Based on the
Bogoliubov approach, expressions for the total- and relative-phase fluctuations
are derived analytically for the low temperature regime. Numerical simulations
of the stochastic projected Gross-Pitaevskii equation (SPGPE) give a good
agreement with the analytical predictions.
