  We study the Uhlmann holonomy [Rep. Math. Phys. 24, 229 (1986)] of quantum
states for hydrogen-like atoms where the intrinsic spin and orbital angular
momentum are coupled by the spin-orbit interaction and subject to a slowly
varying magnetic field. We show that the holonomy for the orbital angular
momentum and spin subsystems is non-Abelian, while the holonomy of the whole
system is Abelian. Quantum entanglement in the states of the whole system is
crucially related to the non-Abelian gauge structure of the subsystems. We
analyze the phase of the Wilson loop variable associated with the Uhlmann
holonomy, and find a relation between the phase of the whole system with
corresponding marginal phases. Based on the result for the model system we
provide evidence that the phase of the Wilson loop variable and the mixed-state
geometric phase [E. Sj\"oqvist {\it et al.} Phys. Rev. Lett. 85, 2845 (2000)]
are in general inequivalent.
