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In this paper, the coupled Rayleigh-Taylor-Kelvin-Helmholtz instability(RTI, KHI and RTKHI, respectively) system is investigated using a multiple-relaxation-time discrete Boltzmann model. Both the morphological boundary length and thermodynamic nonequilibrium (TNE) strength are introduced to probe the complex configurations and kinetic processes. In the simulations, RTI always plays a major role in the later stage, while the main mechanism in the early stage depends on the comparison of buoyancy and shear strength. It is found that, both the total boundary length $L$ of the condensed temperature field and the mean heat flux strength $D_{3,1}$ can be used to measure the ratio of buoyancy to shear strength, and to quantitatively judge the main mechanism in the early stage of the RTKHI system. Specifically, when KHI (RTI) dominates, $L^{KHI} > L^{RTI}$ ($L^{KHI} < L^{RTI}$), $D_{3,1}^{KHI} > D_{3,1}^{RTI}$ ($D_{3,1}^{KHI} < D_{3,1}^{RTI}$); when KHI and RTI are balanced, $L^{KHI} = L^{RTI}$, $D_{3,1}^{KHI} = D_{3,1}^{RTI}$. A second sets of findings are as below: For the case where the KHI dominates at earlier time and the RTI dominates at later time, the evolution process can be roughly divided into two stages. Before the transition point of the two stages, $L^{RTKHI}$ initially increases exponentially, and then increases linearly. Hence, the ending point of linear increasing $L^{RTKHI}$ can work as a geometric criterion for discriminating the two stages. The TNE quantity, heat flux strength $D_{3,1}^{RTKHI}$, shows similar behavior. Therefore, the ending point of linear increasing $D_{3,1}^{RTKHI}$ can work as a physical criterion for discriminating the two stages.