An experimental arrangement is proposed which combines the idea of two-photon interferometry with that of geometric phases. A pair of photons generated by parametric down-conversion is fed into a pair of Mach-Zehnder interferometers, in each of which there is phase shift produced by the helically wound optical fibre in the manner of Tomita and Chiao(1986). If the winding angle is a multiple of 2 pi there is a closed path in parameter space, in accordance with the pioneering analysis of Berry(1984). More general windings are also considered in which the path in parameter space is not closed, and the geometric phase for such open paths is calculated in accordance with the gauge-invariant prescription of Aitchison and Wanelik. If the proposed experiment is realized, it will provide a test for the noncyclic geometric phase for photons and for the set of prescriptions equivalent to that of Aitchison and Wanelik(1992). Furthermore, the geometric phase in this arrangement will be unequivocally quantum mechanical because of the non-classical character of two-photon interferometry.