Take two identical closed strings, both tracing out exactly the same path in space. These two strings are coincident everywhere. Call this state I.
Take a single closed string following exactly the same closed path as in the first case, but not closing just yet. The string goes around exactly the same path once again before closing in on itself. Two cycles around the same closed path. Call this state II.
String field theory tells us unambiguously states I and II are distinct.
Stretch this closed path to make it much larger than the string scale. Supposedly, stringy nonlocality only happens at the string scale. States I and II still differ.
Partition target space into local regions the size of the string scale. The path cuts across a chain of such local regions. If string theory were local, we can reconstruct the state of the entire universe from the restricted states of each subregion if we allow for quantum entanglement between regions. Locally, states I and II ought to be indistinguishable over each local region. For each local region, we always see two string segments passing through it. Thus, states I and II have to be identical?
This can't be. Either string theory is inherently nonlocal over scales much larger than the string scale, or it obeys Maxwell-Boltzmann statistics and not Bose-Einstein statistics.
This isn't some Aharonov-Bohm effect. Even if we include all the local regions in the "interior" of the closed loop, this doesn't change matters the least bit.
PS. Please reread my question more carefully. What you call configuration III is actually my configuration I.
PPS: Let me try to understand your explanation. If we have N coincident strings, or a string which winds round the same loop N times or any other combination in between, this can be described by an $S_N$ discrete gauge symmetry. The conjugacy class of the holonomy of this discrete gauge symmetry around the loop distinguishes between the various combinations. Feel free to correct me if I am wrong. This has the flavor of parastatistics, does it not?