A multivariable measurement error model $AX \approx B$ is considered. Here $A$ and $B$ are input and output matrices of measurements and $X$ is a rectangular matrix of fixed size to be estimated. The errors in $[A,B]$ are row-wise independent, but within each row the errors may be correlated. Some of the columns are observed without errors and the error covariance matrices may differ from row to row. The total covariance structure of the errors is known up to a scalar factor. The fully weighted total least squares estimator of $X$ is studied. We give conditions for asymptotic normality of the estimator, as the number of rows in $A$ is increasing. We provide that the covariance structure of the limiting Gaussian random matrix is nonsingular.