Is there an (interesting) mathematical theory in first order logic that is inconsistent with Peano Arithmetic? There are first order theories that don't entail PA, like Tarski's elementary geometry. 
Is there one that isn't consistent with PA, aka T + NON-PA would be sound? (Ideally it wouldn't be a "pathological" example)
 A: There's always PA+$\neg$Con(PA), but that's a bit pathological. More seriously, the most obvious one to my mind is the theory of the field of real numbers, $Th(\mathbb{R};+,\times)$. Perhaps surprisingly, even though the reals are bigger, their theory is simpler - Tarski showed that it's decidable!. The field of rationals, meanwhile, is undecidable (although the complexity of its $\Sigma_1$ theory is still unknown).
There are other things to look at as well. For example, we could look at the theory of Boolean rings; this contains the sentence "$\forall x(x+x=0)$," so clearly contradicts PA. As a more complicated example, we could look at the theory of the field $(\mathbb{Q}; +,\times)$, or of the ring of "integer parts" of Puiseaux series; this contradicts PA, since even simple instances of the induction scheme fail. (Interestingly, it does satisfy a weak version of induction, namely "open induction"; the study of the model theory of this structure is part of the general model theory of induction principles.)
