This question is motivated by wondering to what extent "natural" theories are linearly ordered (or at least ordered in a directed manner) by their (first-order) arithmetic consequences, in analogy to the phenomenon that "natural" theories seem to be linearly ordered by consistency strength.
In an answer to this question, Joel David Hamkins gave several examples illustrating how these two hierarchies -- arithmetic consequence versus consistency strength - may differ, and indeed his examples also illustrate that the ordering by arithmetic consequence is not linear, even for arguably "natural" theories.
But it so happens that these examples largely involved playing games with consistency statements, so the departure from linearity feels "small". One way of making this a bit more precise is that it seems, as far as I can tell, to still be possible that the partial ordering of theories is still directed, at least for "natural" theories. That is, what I know about the matter is consistent with the idea that there really is one "true arithmetic" with which all of our "natural" or "serious" theories (apart from things like $T + \neg Con(T)$ for various $T$) are consistent.
In order to have a hope of challenging this view, it seems one needs an alternate "coherent" picture of the mathematical universe. The only example that comes to mind for me is $ZF + AD$, which is inconsistent with ZFC, but nonetheless its consistency strength is well-calibrated (and nontrivial), and seems to have some sort of "inner logic" to it which could potentially yield a different picture of the world even at the level of arithmetic. AD paints a different picture of the universe at least if we include non-arithmetic statements in our scope. But I'm not sure it does anything unusual at the level of arithmetic.
Are the arithmetic consequences of $ZF+AD$ consistent with standard theories like $ZFC + L$ for various large cardinal axioms $L$?
Are the arithmetic consequences of $ZF+AD$ implied by large cardinals?
More broadly, is there any good candidate out there for a theory $T$ whose arithmetic consequences are inconsistent with (or at least not implied by) ZFC + large cardinals, such that $T$ has some kind of "inner coherence" (in analogy to how large cardinals are said to have "inner coherence" by virtue of inner model theory -- as opposed to theories of the form $T + \neg Con(T)$ and the like)?