We present a framework to compute non-Gaussian likelihoods for two-point correlation functions. The non-Gaussianity is most pronounced on large scales that will be well-measured by stage-IV weak-lensing surveys. We show how such a multivariate likelihood can be constructed and efficiently evaluated using a copula approach by incorporating exact one-dimensional marginals and a dependence structure derived from the exact multivariate likelihood. The copula likelihood is found to be in better agreement with simulated sampling distributions of correlation functions than Gaussian likelihoods, particularly on large scales. We furthermore investigate the effect of the non-Gaussian copula likelihood on posterior inference, including sampling the full parameter space of contemporary weak-lensing analyses. We find potential parameter shifts in $S_8$ on the order of one standard deviation for $1 \ 000 \ \mathrm{deg}^2$ surveys but negligible shifts for areas of $10 \ 000 \ \mathrm{deg}^2$, suggesting Gaussian likelihoods are sufficient for stage-IV surveys, though results depend on the detailed mask geometry and data-vector structure.

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