The necessary and sufficient condition for existence of a complex square matrix with prescribed (nonnegative) singular values and (complex) eigenvalues (viz., the Weyl-Horn inequalities) is generalized to the case where not all the eigenvalues are prescribed (and, by the way, an amusing observation about the “missing” eigenvalues is made in this context) and then, the construction itself is addressed. The current available algorithms are made more sophisticated - the order of eigenvalues may be prescribed in triangular case. This idea improves the recent algorithm of M. T. Chu [SIAM J. Numer. Anal. 37, No. 3, 1004-1020 (2000; Zbl 0994.65040)] and, as the most important consequence, the matrix may be required real (one first constructs its block-triangular predecessor with the necessary two-by-two partitioning).