We consider variance estimation in the case of a calibrated estimator of a finite population total of a variable of interest Y. An estimator is calibrated with respect to a particular design variable X when the estimator is such as to yield the actual population total of the X's when X is taken as the variable of interest (that is, when it is substituted for Y in the estimation formula), whatever the sample. In this situation, we say a variance estimator is zero-calibrated if, for Y = X, it takes the value 0. This is a desirable property, since otherwise we may suspect the variance estimator of introducing gratuitous noise. We consider the utility of this criterion in identifying undesirable estimators as well as in adjusting weights for replication variance estimators in two-phase sampling.