  Assuming perfect detection efficiency, we present an (indeterministic) model
for an EPR-Bohm experiment which reproduces the singlet correlations, without
contradicting Bell's original locality condition. In this model we allow the
probability distribution $\rho_{\lambda}$ of the state $\lambda$ at the source
to depend parametrically on the orientation $\xi$ of one of the measuring
devices: $\rho_{\lambda}(\lambda,\xi)$. In a Bell experiment, no-signaling
between the source and each one of the devices would seem clearly sufficient to
rule such an influence; however, not even schemes where the choice of
observables takes place during the on-flight time of the particles can prevent,
in some situations, a model of this type from violating the local bounds. In
particular, a random shift $\rho_{\lambda}(\lambda,\xi_1) \rightarrow
\rho_{\lambda}(\lambda,\xi_2) \rightarrow...\rightarrow
\rho_{\lambda}(\lambda,\xi_n)$ allows the model to perform a "subensemble
selection" for each of the terms involved in the inequality (analogous to what
goes on with the efficiency loophole), whenever some correlation of those
$\rho_{\lambda}$-shifts with the sequence of measurement choices is allowed.
That correlation does not necessarily imply signaling during the photon
on-flight time.
