It does, unless you have a way to tune your Hamiltonian such that $\Delta$ becomes zero. Since a tunable Hamiltonian is something you usually want in a quantum computer implementation, this should not be a problem.
If this term is non-switchable, it just means that the basis in which you are working is continuously rotating, and you have to keep track of ...
They are always rotating in the lab reference frame, but most quantum algorithms take a rotating reference frame to simply things, so that a z rotation only happens when you want it to.
The rotating reference frame spins along with the natural spinning rate of each qubit, so in general in can spin at different rates for different qubits.