This only holds if[Edit: there are no rational branch pointswas an obvious mistake in my original answer, e.gwhich was noticed in the comments. ifHere's the cover is étaleamended statement. Over every] 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 otherother; if $P$ is a branch point, then on both $B$ and $B'$ there is a single rational point over $P$.
