This paper presents a drop-threshold incomplete LD-1LT (δ) factorization constraint preconditioner for saddle-point systems using a threshold parameter δ. A transformed saddle-point matrix is partitioned into a block structure with blocks of order 1 and 2 constituting ‘a priori pivots’. Based on these pivots an incomplete LD-1LT (δ) factorization constraint preconditioner is computed that approaches an exact form as δ approaches zero. We prove that both the exact and incomplete factorizations exist such that the entries of the constraint block remain unaltered in the triangular factors. Numerical results are presented for validation.