[Edit: there was an obvious mistake in my original answer, which was noticed in the comments. Here's the amended statement.] Let $P$ be a rational point on $C$, then if $P$ is not a branch point then there are two rational points over $P$ on precisely one of $B$ and $B'$ and zero on the other; if $P$ is a branch point, then on both $B$ and $B'$ there is a single rational point over $P$.
R.P.
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