What are the mathematical problems in introducing Spin 3/2 fermions? Can the physics complications of introducing spin 3/2 Rarita-Schwinger matter be put in geometric (or other) terms readily accessible to a mathematician?
 A: Free spin 3/2 fields cause no problems; see Weinberg's QFT book, Volume 1.
The problem with elementary spin 3/2 fields is the difficulty of accounting for the interaction with the electromagnetic field. The Rarita-Schwinger field equations with the standard minimal coupling via the covariant derivative violate causality, as they allow superluminal signalling - already on the single particle level. 
Nonrenormalizability is another issue, but could be handled in the sense of effective field theories if the other defect were absent.
A: You may be interested in following article: 
Thomas-Paul Hack, Mathias Makedonski "A No-Go Theorem for the Consistent Quantization of the Massive Gravitino on Robertson-Walker Spacetimes and Arbitrary Spin 3/2 Fields on General Curved Spacetimes"
http://arxiv.org/abs/1106.6327
A: Physics complications of "introducing" spin 3/2 matter are the same as for spin 1/2 and spin 0 - the initial approximation in the corresponding interaction theory is physically wrong and calculations give too big (= just wrong) perturbative corrections. It is a complete failure of physics description and it cannot be casted in "geometric terms". Most people, however, does not see it. 
Edit for downvoters: While in case of Rarita-Schwinger equation the solution violates even causality and it cannot be repaired with the constant renormalizations, this feature is still not considered as a failure of coupling. Indeed, we cannot be wrong. It is nature who is wrong, especially at short distances.
