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We determine the charm and strange quark masses in the $\overline{\text{MS}}$scheme, using $n_f=2+1+1$ lattice QCD calculations with highly improvedstaggered quarks (HISQ) and the RI-SMOM intermediate scheme to connect the barelattice quark masses to continuum renormalisation schemes. Our study coversanalysis of systematic uncertainties from this method, includingnonperturbative artefacts and the impact of the non-zero physical sea quarkmasses. We find $m_c^{\overline{\text{MS}}}(3 \text{GeV}) = 0.9896(61)$ GeV and$m_s^{\overline{\text{MS}}}(3 \text{GeV}) = 0.08536(85)$ GeV, where theuncertainties are dominated by the tuning of the bare lattice quark masses.These results are consistent with, and of similar accuracy to, those using thecurrent-current correlator approach coupled to high-order continuum QCDperturbation theory, implemented in the same quark formalism and on the samegauge field configurations. This provides a strong test of the consistency ofmethods for determining the quark masses to high precision from lattice QCD. Wealso give updated lattice QCD world averages for $c$ and $s$ quark masses.

Determination of quark masses from $\mathbf{n_f=4}$ lattice QCD and the RI-SMOM intermediate scheme