This question is about the "old-fashioned" coherent sequences, in the style of Mitchell
Mitchell, W. (1974). Sets constructible from sequences of ultrafilters. Journal of Symbolic Logic, 39(1), 57-66. doi:10.2307/2272343
Recall that a coherent sequence of measures of length $\ell$ is given by a function $o^{\mathcal{U}}\colon \ell\to \mathrm{Ord}$ and a sequence $$\mathcal{U}=\langle U_\alpha^\beta; \alpha<\ell, \beta<o^{\mathcal{U}}(\alpha)\rangle$$ where each $U_\alpha^\beta$ is a normal measure on $\alpha$ and, if $j_\alpha^\beta$ is the associated embedding, $$j_\alpha^\beta(\mathcal{U})\upharpoonright (\alpha+1,0) = \mathcal{U} \upharpoonright (\alpha,\beta)$$ (the restrictions are taken in the sense of the lexicographic order).
Now let's suppose that $V=L[\mathcal{U}]$ believes that $\mathcal{U}$ is a coherent sequence of normal measures, and look at some initial segment $\mathcal{U}\upharpoonright (\delta,0)$ of $\mathcal{U}$. Can we conclude that $\mathcal{U}\upharpoonright (\delta,0)$ is coherent in $L[\mathcal{U}\upharpoonright (\delta,0)]$ (that $\mathcal{U}\upharpoonright (\delta,0)$ is coherent in $V$ follows from the definition, as pointed out in the comments)?
Typically I would want to argue for this by saying that all relevant functions with domain $\gamma<\delta$ in $L[\mathcal{U}]$ are constructed before the next measurable after $\gamma$ appears on the $\mathcal{U}$ sequence, so the ultrapowers of $L[\mathcal{U}]$ and $L[\mathcal{U}\upharpoonright(\delta,0)]$ match well enough for the coherence property to be satisfied, but from what I can tell, $L[\mathcal{U}]$ does not have any of the nice condensation properties that would allow an argument like that to work (and unless I'm mistaken, reals such as $0^\sharp$ will definitely only appear after a measurable stage).
I can also imagine an argument that takes $L[\mathcal{U}]$ and iterates the first measure on $\mathcal{U}$ at or after $\delta$ out of the universe. I think that the limit model of this iteration should just be $L[\mathcal{U}\upharpoonright (\delta,0)]$, and elementarity should show that this initial segment of $\mathcal{U}$ is coherent, but I am not confident.