Constant Regressor in GLS Consider the following regression model:

$y_{i1}=\beta_1 +u_{i1}$
$y_{i2}=\beta_{21}+\beta_{22}x_i+u_{i2}$.

If $E(x_i' u_{i1})\neq 0$ and $E(x_i' u_{i2})=0$, will we get consistent estimators for $\beta_1$ and $\beta_{21}$ using GLS?
I think we will not get consistent estimators for both, because the assumption $E(x_i\otimes u_i)=0$ is violated. Please correct me if I am wrong. Thanks in advance!
The equation-wise OLS estimators are consistent under the further normalization that $E(u_{i1})=0$ and $E(u_{i2})=0$, which are naturally assumed to be satisfied. For GLS, I don't know what you mean by GLS, but if you mean the SUR estimator, then I see no reasons why inconsistent. The fact that $E(x_i' u_{i1}) \ne 0$ is irrelevant because $x_i$ is not a regressor in the first equation.