Why not regard all large gauge transformations as genuine ones? A large gauge transformation is a gauge transformation that is not connected to the identity. When quantizing a gauge theory, we must take configurations related by ordinary gauge transformations to represent the same physical state, but it is ambiguous whether large gauge transformations should be considered as true gauge transformations.
For example, the typical treatment of Yang-Mills compactifies space to $S^3$, and finds several vacua $|n \rangle$ which are related only by large gauge transformations. Since instantons allow tunneling between them, the physical vacuum is a $\theta$-vacuum of the form
$$|\theta \rangle = \sum_n e^{i n \theta} |n \rangle.$$
However, E. Weinberg presents a different view in his book Classical Solutions in Quantum Field Theory, with more detail in this paper. Suppose one works in a gauge where there is a one-to-one correspondence between $F_{\mu\nu}$ and $A_\mu$, e.g.
$$A_3 = 0, \quad A_2|_{z = 0} = 0, \quad A_1|_{y = z = 0}, \quad A_0|_{x = y = z = 0} = 0.$$
There is no gauge freedom here because the first condition leaves only $z$-independent gauge transformations, then the second leaves only $z$-independent and $y$-independent gauge transformations, and so on. Hence there is a unique vacuum, corresponding to $A_\mu = 0$, and there is no such thing as a $\theta$-vacuum. The key here is that establishing this gauge requires large gauge transformations, so Weinberg has implicitly taken them to be do-nothing operations.
Though this formulation is different from the usual one, it seems to give all the same physical predictions. For example, instantons still exist, but they are tunneling events from one vacuum to itself, analogous to a pendulum rotating by a full turn. The observable effects of instantons, such as baryon number violation, hold just as well. The $\theta$-term of QCD need not be induced by the $\theta$ vacuum, but can simply be put into the Lagrangian since it is allowed by symmetries. 
Hence for Yang-Mills, we seem to lose nothing by taking all large gauge transformations to be do-nothing operations, and we gain simplicity and clarity. Are there any downsides? Specifically, is there any measurable quantity that Weinberg's formalism would get wrong, and the more common one would get right? More generally, why don't we always mod out by large gauge transformations?
 A: You cannot generally use large gauge transformations as "do nothing" operations because while they connect classically equivalent systems, the quantizations of these systems may be inequivalent, cf. this answer by David Bar Moshe and references therein. 
In short, the example given there is the Witten effect (named after Witten's description of it in "Dyons of charge $eθ/2π$") producing dyons with fractional electric charge when $\theta$ is non-zero. The dyon appears as "the monopole state" when quantizing the theory modulo small gauge transformations.
On a more abstract level, it is the usual quantization process itself that means that quantumly only the small gauge transformations are guaranteed to "do nothing": Both the Dirac-Bergmann recipe and the BRST formalism focus on the algebra of gauge transformations and enforcing its "do nothing" character on the quantum physical space of states. Nothing in these recipes enforces that large gauge transformations would be "do nothing" operations in the quantum gauge theory, since the algebra can only ever exponentiate to transformations connected to the identity. You might get away with claiming they do, but the standard quantization of gauge systems gives you no solid footing to do so, not even by the laxer standards of physicists.
