The gauge field allows the electro-magnetic field to be calculated, however we cannot calculate the electro-magnetic field from the gauge field. There are many gauge fields that will produce a given electro-magnetic field and there is no way to find which it is.
All forces are now known to be governed by gauge symmetries, descriptions of which are called gauge theories.
The gauge field is the phase of the wave function.
The wave function needs to include the effect of local changes in phase. To do this we add A(x) term, this accounts for the electromagnetic force field.