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.
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