L-moments are useful summary statistics and so have been widely employed in statistical analysis. L-comoments provide a multivariate generalization of L-moments. Joint finite-sample covariance structure, which represents the sampling uncertainty, of L-comoments is less well understood than that of univariate L-moments. We thus derive exact finite-sample covariance formulas for sample L-comoments using explicit conditional U-statistic kernels. We further construct an asymptotic projection estimator and an unbiased estimator of the finite-sample covariance matrix and develop positive-definite corrections. An efficient rank-prefix algorithm evaluates the full unbiased covariance estimator in $O(n^2)$ time without enumerating disjoint subsets. Numerical experiments demonstrate substantial computational savings. The results provide a basis for first-order covariance approximations for L-comoment ratios and for subsequent multivariate L-moment estimation.

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