"I'm having a hard time saying exactly what I mean by chimericity in general, but some non-examples may convey a better sense of what I don't mean by the term.

"A number system consisting of the positive reals and the negative integers would be chimeric, but since it doesn't arise naturally (as far as I know), it doesn't qualify."

That reminds me a little bit of the fact that a Wishart distribution with $n$ degrees of freedom on $p\times p$ nonnegative-definite symmetric matrices exists precisely if $n \in \lbrace 0, \dots , p-1 \rbrace \cup (p-1,\infty)$. The sum of two $p\times p$ independent Wishart-distributed random matrices with respective degrees of freedom $n_1$ and $n_2$ has a Wishart distribution with $n_1+n_2$ degrees of freedom, so the operation of addition on this somewhat odd-looking set matters.