supervaluationist structuralism
Supervaluationsism and structuralism ('eliminative structuralism', not the kind of structuralism that postulates structures) almost coincide. Structuralism about something t says that any sentence 'F(t)' is to be interpreted as 'for all x (if x is a candidate for t then F(x))'. For example, arithmetical structuralism says that '2+2=4' is to be interpreted as 'for all [N,0,',+,.] (if [N,0,',+,.] satisfies the peano axioms then 0''+0''=0''''). If we translate 'x is a candidate for t' as 'x is the referent of a precisification of 't'', we get: F(t) is (super-)true iff it is true on all precisifications of 't'.
Difference between structuralism and supervaluationsism comes in the treatment of falsity. For structuralism, F(t) is (super-)false iff it is false on any precisification. For supervaluationism, it is false iff it is false on all precisifications. Assume a mathematical statement G(t) is true on some but not all precisifications. That is, there is some candidate x for t such that G(x) and some other candidate x such that not G(x). So being a candidate does not decide whether G applies. In other words, G(t) is formally undecidable.
Now either the conditions for being a candidate are complete or not. If they are complete, then there are no formally undecidable sentences, and structuralism and supervaluationsism coincide. If they aren't complete, then on the structuralist account, undecidable sentences are all false, whereas on the supervaluationist account, they are neither true nor false. Supervaluationism appears clearly superior here.
I think it is also superior in another respect: Supervaluationists usually don't claim that vague statements are somehow really quantifications over (the referents of) precisifications. So giving a supervaluationist account of mathematics -- treating mathematical notions as indeterminate -- looks less revisionary. It is even compatible with intuitions that these notions are not indeterminate, for indeterminacy can be unintended and surprising, as witness the 'jade' case.