Can one find two polynomials $P,Q \in \mathbb{R}[X,Y]$ with the same non-empty zero set, that have no common factors ?
If we took complex coefficients instead of real coefficients, the answer would be no (see here).
Note that $P=X^2+1$ and $Q=Y^2+1$ have no common factors, and the same zero set $\emptyset$. I'm looking for a less trivial example.
EDIT: Is it possible to find an example where the zero set is an infinite set ?
Let $A,B \in \mathbb{R}[X,Y]$ and $P_{a,b} = a A^2 + bB^2$. Then the family of polynomials $\{P_{a,b} \mid a,b \in \mathbb{R}^+\}\,$ all have the same zero set $\{(x,y) \mid A(x,y) = 0\} \cap \{(x,y) \mid B(x,y) = 0\} $.
[ EDIT ] To cover the coprimality question asked in a comment, take for example $A,B$ to have no common factors, then $P_{a,b}$ and $P_{a,2b}$ have no common factors, either.