  The recently reported violation of a Bell inequality using entangled
electronic spins in diamonds (Hensen et al., Nature 526, 682-686) provided the
first loophole-free evidence against local-realist theories of nature. Here we
report on data from a second Bell experiment using the same experimental setup
with minor modifications. We find a violation of the CHSH-Bell inequality of
$2.35 \pm 0.18$, in agreement with the first run, yielding an overall value of
$S = 2.38 \pm 0.14$. We calculate the resulting $P$-values of the second
experiment and of the combined Bell tests. We provide an additional analysis of
the distribution of settings choices recorded during the two tests, finding
that the observed distributions are consistent with uniform settings for both
tests. Finally, we analytically study the effect of particular models of random
number generator (RNG) imperfection on our hypothesis test. We find that the
winning probability per trial in the CHSH game can be bounded knowing only the
mean of the RNG bias, implying that our experimental result is robust for any
model underlying the estimated average RNG bias.
