Binaries in which both stars are pulsating are rare but extremely valuable. We present the first study of an asteroseismic binary system consisting of a core helium-burning red clump (RC) star and a red giant branch (RGB) star. The Kepler target KIC 10841730 is a wide binary (period $2917 \pm 8$ d) that provides ideal conditions to test the accuracy of RC models. While prior studies of RC stars have revealed discrepancies in modelling the period spacings of mixed modes, other model parameters remain largely untested. We perform a detailed modelling analysis using individual mode frequencies and cover a large parameter space in mass, metallicity, He-abundance, mixing length, overshooting, and mass-loss, and we also explore different methods to correct for surface effects. We find two possible results for the RC models. One solution requires introducing an unexpected offset of the phase shift in the RC model, yielding an age consistent with the companion star and current masses of $1.01\pm 0.06$ and $1.08\pm 0.06$ M$_{\mathrm{\odot }}$ for the RC and RGB star, respectively. Alternatively, we find that excluding the identification of two questionable radial modes resolves the phase-shift offset issue but results in a higher mass and thus a much younger age for the RC star, contradicting the age obtained from its companion. We conclude that uncertainties in RC models affect not only the g-mode period spacings but also the properties of the p modes. We show the power of asteroseismic binaries in validating and constraining stellar models and highlight the need for refining RC models.