A plausible formation scenario for the Galactic globular clusters 47 Tucanae (47 Tuc) and Omega Centauri $(\omega$ Cen) is that they are tidally stripped remnants of dwarf galaxies, in which case they are likely to have retained a fraction of their dark matter cores. In this study, we have used the ultra-wide band receiver on the Parkes telescope (Murriyang) to place upper limits on the annihilation rate of exotic Light Dark Matter particles $(\chi)$ via the $\chi\chi\rightarrow e^+e^-$ channel using measurements of the recombination rate of positronium (Ps). This is an extension of a technique previously used to search for Ps in the Galactic Centre. However, by stacking of spectral data at multiple line frequencies, we have been able to improve sensitivity. Our measurements have resulted in $3-\sigma$ flux density (recombination rate) upper limits of 1.7 mJy $\left(1.4\times 10^{43}\, \mathrm{s}^{-1}\right)$ and 0.8 mJy $\left(1.1 \times 10^{43} \mathrm{s}^{-1}\right)$ for 47 Tuc and $\omega$ Cen, respectively. Within the Parkes beam at the cluster distances, which varies from 10–23 pc depending on the frequency of the recombination line, and for an assumed annihilation cross-section $\langle\sigma v\rangle = 3\times 10^{-29} \mathrm{cm}^3\, \mathrm{s}^{-1}$ , we calculate upper limits to the dark matter mass and rms dark matter density of ${\lesssim} 1.2-1.3\times 10^5 f_n^{-0.5}$ $\left(m_\chi/\mathrm{MeV\, c}^{-2}\right)$ $\mathrm{M}_{\odot}$ and ${\lesssim} 48-54 f_n^{-0.5}$ $\left(m_\chi/\mathrm{MeV\, c}^{-2}\right)$ $\mathrm{M}_{\odot} \mathrm{pc}^{-3}$ for the clusters, where $f_n=R_n/R_p$ is the ratio of Ps recombination transitions to annihilations, estimated to be ${\sim}0.01$ . The radio limits for $\omega$ Cen suggest that, for a fiducial dark/luminous mass ratio of ${\sim}0.05$ , any contribution from Light Dark Matter is small unless $\langle\sigma v\rangle < 7.9\times 10^{-28}\ \left(m_\chi/\mathrm{MeV\, c}^{-2}\right)^2 \mathrm{cm}^3 \mathrm{s}^{-1}$ . Owing to the compactness and proximity of the clusters, archival 511-keV measurements suggest even tighter limits than permitted by CMB anisotropies, $\langle\sigma v\rangle < 8.6\times 10^{-31}\ (m_\chi/\mathrm{MeV\, c}^{-2})^2 \mathrm{cm}^3 \mathrm{s}^{-1}$ . Due to the very low synchrotron radiation background, our recombination rate limits substantially improve on previous radio limits for the Milky Way.