Suppose you are in radial free fall at some point outside the event horizon of a Schwarzchild metric. The strong equivalence principle implies that locally you would be unable to discern whether you are in fact in free fall near a gravitating body or simply at rest in flat spacetime. The operative word being "locally." If you had, say, two bowling balls freely falling with you nearby, then (due to differences in the gravitational field) you would observe them gradual approach one another - an effect called geodesic deviation.
This phenomenon is usually cited as an example of the inexact nature of the equivalence between gravity and acceleration or of the local nature of the equivalence principle, but it seems to me that (at least under certain circumstances) this equivalence can be made truly exact by appealing to other explanations for the observations. For example, in the situation described above, the two bowling balls would move together even in nominally flat spacetime due to their mutual gravitational attraction. Granted, they may come together more rapidly in the freely falling situation due to the addition of tidal forces, but without any other data, one might simply conclude that the bowling balls have more mass than they actually do - just enough to account for the extra attraction. Or, alternatively, one might do some experiment to determine the inertial mass of the bowling balls and then use their mutual attraction in the freely falling frame to determine the gravitational constant.
Even if these mental maneuvers work as explanations, it is fairly easy to come up with examples that would seem to undermine an attempt to explain away tidal effects completely. For example, suppose you are in an unstable circular orbit with one bowling ball at a slightly smaller radius and the other at a slightly larger radius. Over time, the three of you would diverge - one spiraling inward, one gradually going out to infinity, and you staying in your circular orbit. Without some profound changes in one's theory of gravity, this observation seems impossible to square with flat spacetime.
On the other hand, both of these examples would require very careful preparation and very delicate measurements if they were to actually be performed, so it doesn't seem entirely implausible that the subtlety of the effects being measured would leave room for apparently minor details that I'm ignoring to change the outcome of the experiments - either leaving the door open to plausible explanations consistent with flat spacetime or providing an additional piece of information to distinguish between them.
Are there any situations in which the equivalence principle can be strengthened such that a freely falling observer under these specific conditions is entirely unable to determine whether or not they are in the presence of a gravitating body? If so, what would the necessary conditions be?