We study the properties of Bose-Einstein condensates under a non-Hermitian spin-orbit coupling (SOC),induced by a dissipative two-photon Raman process.We focus on the dynamics of the condensate at short times,when the impact of decoherence induced by quantum jumps is negligible and the dynamics is coherently driven by a non-Hermitian Hamil-tonian.Given the significantly modified single-particle physics by dissipative SOC,the interplay of non-Hermiticity and interaction leads to a quasi-steady-state phase diagram different from its Hermitian counterpart.In particular,we find that dissipation can induce a phase transition from the stripe phase to the plane-wave phase.We further map out the phase diagram with respect to the dissipation and interaction strengths,and finally investigate the stability of quasi-steady states through the time-dependent dissipative Gross-Pitaevskii equation.Our results are readily accessible based on standard experiments with synthetic spin-orbit couplings.