In this work we apply the line-integral technique to study possible geometric phase effects in the 2 x 2 diabatic double many-body expansion (DMBE) potential energy surface of three hydrogenic systems, namely, H-3, DH2, and HD2. First, we show that the phase obtained by employing the line-integral method is identical (up to a constant) to the ordinary diabatic angle of the orthogonal transformation that diagonalizes the diabatic potential matrix. Next this angle is studied numerically along the line formed by fixing the two hyperspherical coordinates rho and theta and letting Phi change along the interval [0, 2 pi]. We find that in the H-3 system, where this line always encircles the seam, the corresponding line integral always produces the value pi for the geometric (Berry) phase. In the cases of the two isotopic systems we usually find the same results, but we also verify that for substantial regions in configuration space these lines do not encircle the seam and that, therefore, the line integrals produce the value of zero for the geometric phase. Analyzing the results, we establish that the Longuet-Higgins phase, which is usually assumed to be equal to Phi/2, is in general significantly different from this value for all studied mass combinations.