  We study the effect of a magnetic field on the low energy description of Mott
insulators with strong spin-orbit (SO) coupling. In contrast to the standard
case of the Hubbard model without SO coupling, we show that Peierls phases can
modulate the magnetic exchange at leading order in the interaction. Our
mechanism crucially depends on the existence of distinct exchange paths between
neighboring magnetic ions enclosing a well-defined area. Thus it will
generically be present in any solid state realisation of the Kitaev model and
its extensions. We explicitly calculate the variation of the exchange constants
of the so-called $JK\Gamma$ model as a function of the magnetic flux. We
discuss experimental implications of our findings for various settings of
candidate Kitaev spin liquids.
