Interpreting a solution of the Einstein equations for a given matter content I've been always confused with the way people interpret solutions of Einstein equations for some given source. 
Here's an example:
Given some particular solution $(g,F)$ of the Einstein-Maxwell equations, I saw people say stuff like
"this solution describes electromagnetic wave propagating in the AdS background". 
But is it actually possible to determine what exactly the background metric, in which the elmag wave propagates, is? 
It is also question how precisely is the notion of "background metric" defined in such cases - is it defined just by putting $F=0$?
This does not seem right, since $g$ could be then any vacuum spacetime.
EDIT: It seems that such interpretations are always tied to a specific coordinate form of the solution in question. For example, if the line element of the solution can be written in the form 
$$\{\text{some known vacuum } ds^2\} + \{\text{additional terms ammounting to the presence of the electromagnetic field}\},$$ then it is reasonable to conclude that such solution represents an electromagnetic wave propagating in the known background spacetime with $ds^2$. Right?
 A: Generally, it is difficult to unambiguously separate background spacetime from the full solution. However, when people use phrases like "solution will describe a <...> field propagating in <...> spacetime", what they really are talking about is a family of solutions $(g_\alpha,F_\alpha)$ parametrized by some set of variables. This $\alpha$ could be a finite set of real values or it could contain arbitrary functions of several arguments (like in pp-wave solution, for example).
The background spacetime would be a solution corresponding to specific $\alpha$ which would then possess some special properties: enhanced symmetries, absence of singularities, etc. For solutions where there are EM waves propagating, the condition selecting background could be the vanishing of stress-energy tensor.
OP's example where the metric contains separate terms corresponding to EM field is indeed one case where it is quite easy to extract the background metric. However since EFE are nonlinear, generally one cannot expect that background part of the metric would be present as a separate term. Moreover, in some cases  this background would be some sort of singular limit of a generic solution.
