This is a refinement of my question asked earlier, which is answered beautifully in the negative by Thomas Browning. The example he gave was geometrically reducible. Now I want to ask the same question, but with the extra assumptions that $f(0,0) = 0$ and $f$ is geometrically irreducible for all $c \in \mathbb{R}$.

For convenience, the question is: does there exist a polynomial $f \in \mathbb{Z}[x,y]$ such that $f(0,0) = 0$ and $f$ is geometrically irreducible, and such that

$$\displaystyle f(a,b) > 0 \text{ for all } a,b \in \mathbb{Z}$$

and

$$\displaystyle \liminf_{\mathbb{R}^2} f(x,y) = -\infty?$$