One often suggested "route" for the Riemann Hypothesis is through the study of quasicrystals. In particular, assuming RH, the nontrivial zeroes form a 1-dimensional quasicrystal. If we can classify all the 1d quasicrystals then it might be possible to determine which one would correspond to the nontrivial zeroes of $\zeta$ assuming RH, and then perhaps from there we can prove that the particular quasicrystal in question either does or doesn't have the required connection to primes for RH, therefore solving the problem.
However, there could also be a sort of 'inverse' version of the above route. What if instead of describing a rigid structure on the critical line zeroes assuming RH, we described a rigid structure on the counterexample zeroes assuming the failure of RH?
We already know that the counterexample zeroes must come in 4s; every zero has a corresponding zero when reflected across the critical line or the real axis.
One possibility I've played around with is that the "pairs of 4" phenomenon in the counterexample zeroes might actually just be a slice of some broader set of symmetries on the counterexample zeroes, which can then be realized as a group (or another structure of symmetries; perhaps a groupoid or semigroup). In the (admittedly small) chance that this idea has any solid grounds, contradictions may be derived from asserting certain properties of the structure, which may be seen as different "versions" of the way RH can fail.
Another possibility is that there is a (possibly unary) operation on the counterexample zeroes which could be used to generate new counterexample zeroes in a natural-ish way. Then, by studying properties of this algebra, perhaps it could be shown that e.g. it is at least finitely generated, which would be some progress towards RH, or maybe even rule out possible numbers for the zeroes, e.g. "there cannot be exactly 28 counterexamples to RH." If this approach would work then certainly I'd think it would be very useful.
One last example of what this might look like is that all of the nontrivial zeroes form a particular structure, and from that structure you may be able to look at the particular section which would correspond to zeroes of the critical line. This seems the most plausible of the three examples I gave, however it's not as obvious how it might be fruitful.
As far as I'm aware however, there are no further ideas beyond coming in 4s about what structures could be imposed on the counterexample zeroes.
So, is something like this even possible? My guess would be that it isn't, since the nature of the counterexample zeroes is seemingly as elusive as their existence themselves. I am expecting negative answers to this question, but it would be nice to include in those answers more in-depth explanations of why this approach might be likely to fail.
Edit: It has come to my attention that there is some skepticism about the validity of the quasicrystal argument suggested by Freedman. In particular, the quasicrystal community doesn't have a single official definition of "quasicrystal" (yet?) but every definition they've made requires necessarily that a quasicrystal be a Delone set, which, even under RH, the nontrivial zeroes of the RZF are obviously not.