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In this paper, we study how to apply a periodic driving field to control stable spin tunneling in a non-Hermitian spin-orbit coupled bosonic double-well system. By means of a high-frequency approximation, we obtain the analytical Floquet solutions and their associated quasienergies and thus construct the general non-Floquet solutions of the dissipative spin-orbit coupled bosonic system. Based on detailed analysis of the Floquet quasienergy spectrum, the profound effect of system parameters and the periodic driving field on the stability of spin-dependent tunneling is investigated analytically and numerically for both balanced and unbalanced gain-loss between two wells. Under balanced gain and loss, we find that the stable spin-flipping tunneling is preferentially suppressed with the increase of gain-loss strength. When the ratio of Zeeman field strength to periodic driving frequency $Ω/ω$ is even, there is a possibility that \emph{continuous} stable parameter regions will exist. When $Ω/ω$ is odd, nevertheless, only \emph{discrete} stable parameter regions are found. Under unbalanced gain and loss, whether $Ω/ω$ is even or odd, we can get parametric equilibrium conditions for the existence of stable spin tunneling. The results could be useful for the experiments of controlling stable spin transportation in a non-Hermitian spin-orbit coupled system.