Any double cover of ${\bf RP}^2$ whose branch locus has no real component should do. Even the two-sheeted hyperboloid $y^2 = x_0^2 + x_1^2 + x_2^2$ seems to work.

[Corrected]

Any double cover of ${\bf RP}^2$ whose branch locus has degree $4n$ and no real component should do. An example is the surface $y^2 = x_0^4 + x_1^4 + x_2^4$ in the weighted projective space whose coordinates $(x_0:x_1:x_2::y)$ have degrees $1,1,1,2$. Thus $(x_0:x_1:x_2::y)$ is equivalent to $(\lambda x_0:\lambda x_1:\lambda x_2::\lambda^2 y)$ for nonzero $\lambda \in \bf R$, so there are well-defined components $y>0$ and $y<0$, and the projection to $(x_0:x_1:x_2)$ maps each component homeomorphically to ${\bf RP}^2$.

[Earlier I rashly omitted the "degree $4n$" condition and proposed the two-sheeted hyperboloid $y^2 = x_0^2 + x_1^2 + x_2^2$ as an example. That's clearly wrong because it's isomorphic (as a surface in ${\bf RP}^3$) with a Euclidean sphere.]