  We present the topology of spin-split Fermi surface of CaAgAs as determined
by de Haas-van Alphen (dHvA) effect measurements combined with ab initio
calculations. We have determined the torus-shaped nodal-line Fermi surface from
the dHvA oscillations of $\beta$ and $\gamma$ orbits. The former orbit
encircles the nodal-line, while the latter does not. Nevertheless, a nontrivial
Berry phase is found for both orbits. The nontrivial phase of $\beta$ arises
from the orbital characters, which can be expressed as a pseudospin rotating
around the nodal-line. On the other hand, the phase of $\gamma$ is attributed
to the vortex of real spin texture induced by an antisymmetric spin-orbit
interaction. Our result demonstrates that both the real- and pseudo-spin
textures are indispensable in interpreting the electronic topology in
noncentrosymmetric nodal-line semimetals.
