The Gauss-Faraday law can be viewed the integrability conditions for the existence of the electro-magnetic $4$-potential $A_{\mu}$ via [Poincare Lemma](https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms). 

Or conversely, assuming a globally defined electro-magnetic $4$-potential $A_{\mu}$, the Gauss-Faraday law is trivially satisfied. See also [this](https://physics.stackexchange.com/q/71611/2451) related Phys.SE post.