Assuming a standard Maxwellian for the WIMP velocity distribution, we obtain the bounds from null WIMP search results of 59.5 days of COSINE-100 data on the DAMA/LIBRA-phase2 modulation effect within the context of the non-relativistic effective theory of WIMP-nucleus scattering. Here, we systematically assume that one of the effective operators allowed by Galilean invariance dominates in the effective Hamiltonian of a spin-1/2 dark matter (DM) particle. We find that, although DAMA/LIBRA and COSINE-100 use the same sodium-iodide target, the comparison of the two results still depends on the particle-physics model. This is mainly due to two reasons: i) the WIMP signal spectral shape; ii) the expected modulation fractions, when the upper bound on the time-averaged rate in COSINE-100 is converted into a constraint on the annual modulation component in DAMA/LIBRA . We find that the latter effect is the dominant one. For several effective operators the expected modulation fractions are larger than in the standard spin-independent or spin-dependent interaction cases. As a consequence, compatibility between the modulation effect observed in DAMA/LIBRA and the null result from COSINE-100 is still possible for several non-relativistic operators. At low WIMP masses such relatively high values of the modulation fractions arise because COSINE-100 is mainly sensitive to WIMP-sodium scattering events, due to the higher threshold compared to DAMA/LIBRA . A next COSINE analysis is expected to have a full sensitivity for the 5 σ region of DAMA/LIBRA.