Field measurements in QFT

If I understand it correctly, QFT assigns a field operator to every point in space. For a free scalar field:
$$ hat{phi}(mathbf{x},t)=int frac{d^3p}{(2pi)^3} frac{1}{sqrt{2omega_{mathbf{p}}}} left[hat{a}_{mathbf{p}}
exp(-iomega_{mathbf{p}}t+i{mathbf{p}}cdotmathbf{x}) + hat{a}_{mathbf{p}}^{dagger}
exp(iomega_{mathbf{p}}t-i{mathbf{p}}cdotmathbf{x})right] . $$

This operator is Hermitian, so it corresponds to an observable (which makes sense, as it should be possible to measure the "strength" of the field at every point). A given quantum state can be represented as a superposition of the field operator eigenstates, and measuring the field at point $x$ should collapse the state to one of these eigenstates. My question is: what would happen if the field is measured simultaneously at two different points, $mathbf{x}_1$ and $mathbf{x}_2$? The field operators at these points are different (since they depend on $x$), so the eigenstate decompositions are different. Which eigenstate could the quantum state collapse to, if there are no common eigenstates between the operators at $mathbf{x}_1$ and $mathbf{x}_2$?