Let me say first, that I really like this question. Very unusual question about such well known things (in fact I did not know even that the real quintic $\sum_i x_i^5=0$ is $\mathbb RP^3$).

This is not an answer, but more like my interpretation of this question (hopefully correct one). The first comment is about vanishing cycles. Usually, when we speak about vanishing cycles in symplectic geometry, we speak about lagrangan spheres in a fiber of a Lefshetz fibraton constructed using parallel transport given by the symplectic connection on the fibers. But Lefshetz pencil is just the most simple object of algebraic geometry. We can consider instead other one-parameter families of algebraic varieties, that can have singularities more complicated than double points $A_1$. In this case again we can ask what will be the shape of the vanishing cycle constructed via symplectic parallel transport? How does it look like? The point that I want to make is that in a large case of situations, this vanishing cycle will not look at all like a manifold.

Namely, we will consider the example coming from a function on $\mathbb C^n$ with isolated singularity. I.e., we have an analytic function $F:\mathbb C^n\to \mathbb C$ with isolated singularity at $0$ and consider its level sets: hyper-surfaces $F_t:=F^{-1}(t)$ intersected with $B_\varepsilon(0)$ (the ball of radius $\varepsilon$). Then we know that the homotopy type of $F_t\cap B_{\varepsilon}$ is a bucket of $k$ spheres, where $k$ is the Milnor number of the singularity. Now, I think (and here I can not provide the proof), that the vanishing cycle is a deformation retract of $F_t\cap B_{\varepsilon}$. Hence it can not be diffeomorphic to a manifold unless $k=1$. In this case it has to be homotopy equivalent to a sphere, i.e., homeomoerphic to it (by Poincare).

In the case when the singularity of $F$ is non-degenerate (double point), the vanishing cycle is of course diffeomorphic to $S^n$, but maybe if we consider Brieskorn singularities we will be able to get vanishing cycles that are exotic spheres as well?

The above situation concerns the case when the total space of the fibration is smooth. In this case we does not seem to be able to get $\mathbb RP^{n-1}$ as a vanishing cycle. But if we allow singularities in the total space of the fibration, we can get it. Indeed if we consider the function $\sum_i x_i^2=0$ on the total space $\mathbb C^n/\mathbb Z_2$, where $\mathbb Z_2$ is acting by $z\to -z$, the vanishing cycle will be diffomorphic to $\mathbb RP^{n-1}$. In sum, it seems to me that the following question is interesting: can we describe the class of manifolds what appear as vanishing cycles of algebraic one-parameter families?

As for the question itself, I don't know what is the answer...