I disagree with your premise that fields and curvature are different at all.
The gravitational field strength tensor is (or, can be seen as, but usually isn't) the Riemann curvature tensor of spacetime.
Likewise, the electromagnetic field strength tensor is the curvature tensor of a gauge principal bundle.
The fields and the curvatures are not distinct objects, and one cannot meaningfully talk about which of them is fundamental, and which of them is derived. By gauge theory, unification in the sense of viewing both gravity and all other forces coming from connections and their induced curvatures on bundles has already been achieved. The problem is that gravitational theories are non-renormalizable, and thus not as easily incoporated into the QFT framework as gauge theories, who presume a fixed (Minkowski) metric on spacetime.
However, as the electromagnetic field inherently contains energy/momentum, it is, by Einstein's famous formula, inherently equivalent to mass, so, indeed, the presence of electromagnetic fields will curve spacetime. There is a portion of the stress-energy tensor due to electromagnetic fields. However, this is not all the EM field does, as it acts on charged bodies as something which is not the curvature of spacetime, but the curvature of the gauge bundle.