The following problem appeared in the Kömal (Hungarian mathematical journal) in year 1997. Official Source: Problem N 145 at the end of the page here.
Problem. Does there exist a polynomial $f(x, y, z)$ of real coefficients such that $f(x,y,z)$ is positive if and only if $|x|, |y|, |z|$ are sides of a triangle?
The given inequalities $|x|+|y|>|z|$, $|y|+|z|>|x|$, and $|x|+|z|>|y|$ impose an interesting geometric region in $\mathbb{R}^3$:
I am guessing that the question has a negative answer -- no such polynomial should exist. Any thoughts / comments / hints are appreciated!
Note that either all $|x|+|y|-|z|$, $|y|+|z|-|x|$, and $|z|+|x|-|y|$ are positive or just one of them is negative. Hence,
$$(|x|+|y|-|z|)(|y|+|z|-|x|)(|z|+|x|-|y|)>0 \iff |x|,|y|,|z| \text{ are sides of a triangle.}$$
So, $f^*(x,y,z)=(|x|+|y|-|z|)(|y|+|z|-|x|)(|z|+|x|-|y|)$ can work as a source for generating a desired function. As @Macavity suggested in the comments, we have:
$$f(x,y,z)=(|x|+|y|-|z|)(|y|+|z|-|x|)(|z|+|x|-|y|)(|x|+|y|+|z|) \\=2x^2y^2+2x^2z^2+2y^2z^2-x^4-y^4-z^4.$$
Thus, such a polynomial exists as shown above.
PS: It seems that the OP has been indirectly answered at this link.