Strong Angular-Momentum-Coupling Dependence of Atomic Decay Angular Distributions in a Hybrid $\mathbf{LS/LSJ}$ Framework
State production and its subsequent decay can each require a different angular-momentum coupling treatment. Although long recognized, this distinction is often overlooked in the two-step model, where $LS$ coupling is traditionally applied to both production and decay ($LS/LS$), even when the decay requires $LSJ$ coupling. We introduce a hybrid $LS/LSJ$ formulation that uses established angular-momentum recoupling theory to project the initial $LS$ alignment onto the fine-structure $J$ states, evaluates their decay in $LSJ$ coupling, and, for well-separated $J$ levels, sums their contributions incoherently. The $1s2s2p\,{}^{4}\!P_J$ manifold produced by single-electron capture in $\mathrm{C}^{4+}(1s2s\,{}^{3}\!S)+\mathrm{He}$ collisions provides a good test case, with $M_L$-resolved theoretical production cross sections and absolute measurements of the Auger single-differential cross section available. Using the same $LS$ production cross sections in both treatments, we find that the hybrid $LS/LSJ$ treatment strongly suppresses and can even invert the Auger anisotropy relative to the traditional $LS/LS$ treatment, substantially improving agreement with the measurements. Such an incoherent hybrid formulation provides the appropriate framework when $LS$ coupling governs the initial interaction dynamics while well-separated fine-structure levels require an $LSJ$ description of the decay.