We report on the first extraction of interference fragmentation functions from the semi-inclusive production of two hadron pairs in back-to-back jets in e+e- annihilation. A nonzero asymmetry in the correlation of azimuthal orientations of opposite pi+pi- pairs is related to the transverse polarization of fragmenting quarks through a significant polarized dihadron fragmentation function. Extraction of the latter requires the knowledge of its unpolarized counterpart, the probability density for a quark to fragment in a pi+pi- pair. Since data for the unpolarized cross section are missing, we extract the unpolarized dihadron fragmentation function from a Monte Carlo simulation of the cross section.