  Quantum theory allows for correlations between the outcomes of distant
measurements that are inconsistent with any locally causal theory, as
demonstrated by the violation of a Bell inequality. Typical demonstrations of
these correlations require careful alignment between the measurements, which
requires distant parties to share a reference frame. Here, we prove, following
a numerical observation by Shadbolt et al., that if two parties share a Bell
state and each party randomly chooses three orthogonal measurements, then the
parties will always violate a Bell inequality. Furthermore, we prove that this
probability is highly robust against local depolarizing noise, in that small
levels of noise only decrease the probability of violating a Bell inequality by
a small amount. We also show that generalizing to N parties increases the
robustness against noise. These results improve on previous ones that only
allowed a high probability of violating a Bell inequality for large numbers of
parties.
