If we measure whether a particle has propagated, or an antiparticle has propagated, then there is a finite probability for propagation over spacelike separations. See Weinberg's Gravitation and Cosmology (Ch. 2 Sec. 13). In quantum field theory, in the context of S-matrix, measurements are made at spacial infinity from the interaction region whereas, for example, in neutrino oscillation experiments, the situation is very different. The source-detector distances can be small (17.6 m for KARMEN, and 30.0 m for LSND). When one looks at Weinberg's argument one finds that there is a non-zero probability for spacelike separations. But he dismisses it on the ground that for a proton such a distance is only 10^{-14} cm. But if one considers a particle of neV mass, then one gets a macroscopically large distance ~ 100 m! Very relevant for the LSND-KARMEN anomaly.
In the calculation of the propagator there are two terms: t > t' term and the other t<t' term -- when added together they cancel out for outside the light cone. But in LSND and KARMEN situations only one of the two terms are operative -- for S-matrix calculations both terms are operative.
Also, see Tony Zee's Quantum field theory in a nutshell, pp. 23 and 24 where he too argues for a propagation outside the light cone. In particular, theta(0) -- i.e. t=t' ignored by many authors.