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Two-particle azimuthal correlations have been measured in neutral current deep inelastic ep scattering with virtuality $Q^2> 5$ GeV$^2$ at a centre-of-mass energy $\sqrt{𝑠}= 318$ GeV recorded with the ZEUS detector at HERA. The correlations of charged particles have been measured in the range of laboratory pseudorapidity $−1.5 < \eta < 2.0$ and transverse momentum $0.1 < p_T< 5.0$ GeV and event multiplicities $N_{ch}$ up to six times larger than the average $\langle N_{ch} \rangle ≈ 5$. The two-particle correlations have been measured in terms of the angular observables $c_n{2} = 〈〈\cos nΔφ〉〉$, where $n$ is between 1 and 4 and $∆φ$ is the relative azimuthal angle between the two particles. Comparisons with available models of deep inelastic scattering, which are tuned to reproduce inclusive particle production, suggest that the measured two-particle correlations are dominated by contributions from multijet production. The dependence of the correlations as a function of $Q^2$ has also been studied as well as the correlations in photoproduction events ($Q^2 \approx 0$). The correlations observed here do not indicate the kind of collective behaviour recently observed at the highest RHIC and LHC energies in high-multiplicity hadronic collisions.