Focusing on smooth constrained optimization problems, and inspired by the complementary approximate Karush-Kuhn-Tucker (CAKKT) conditions, this work introduces the weighted complementary Approximate Karush-Kuhn-Tucker (WCAKKT) conditions. They are shown to be verified not only by safeguarded augmented Lagrangian methods, but also by inexact restoration methods, inverse and logarithmic barrier methods, and a penalized algorithm for constrained nonsmooth optimization. Under the analyticity of the feasible set description, and resting upon a desingularization result, the new conditions are proved to be equivalent to the CAKKT conditions. The WCAKKT conditions capture the algebraic elements of the desingularization result needed to characterize CAKKT sequences by means of a weighted complementarity condition that sums zero asymptotically. Due to its generality and strength, the new condition may help to enlighten the practical performance of algorithms in generating CAKKT sequences.