The examples in Ruan's paper "Symplectic topology on algebraic 3-folds" (JDG 1994) seem to qualify: take any two algebraic surfaces V and W which are homeomorphic but such that V is minimal and W isn't. These are nondiffeomorphic, but VxS2 and WxS2 are diffeomorphic, and Ruan gives lots of examples (starting with V equal to the Barlow surface and W equal to the 8-point blowup of CP2) where the diffeomorphism can be arranged to intertwine the first Chern classes, whence by a theorem of Wall the almost complex structures are isotopic. However, the distinction between the GW invariants between V and W (which holds because V is minimal and W isn't) survives to VxS2 and WxS2, so VxS2 and WxS2 aren't symplectic deformation equivalent.
Gromov--Witten invariants are designed to count the "number" of curves in a space in a deformation invariant way. Since the number of curves can change under deformations, the Gromov--Witten invariants won't have a direct interpretation in terms of actual numbers of curves, even taking automorphisms into account.
Here is an example of how a negative number might come up, though strictly speaking it isn't a Gromov--Witten invariant. Let M be the moduli space of maps from P^1 to a the total space of O(-4) on P^1. Call this space X. Note that I said maps from P^1, not a genus zero curve, so the source curve is rigid. That's why this isn't Gromov--Witten theory. Any such map factors through the zero section (since O(-4) has no nonzero sections), so this space is the same as the space of maps from P^1 to itself. I just want to look at degree one maps, so the moduli space is 3 dimensional.
We could also compute the dimension using deformation theory: the deformations of a map f are classified by $H^0(f^\ast T)$ where T is the tangent bundle of the target. The target in this case is O(-4), not just P^1, and the tangent bundle restricts to O(2) + O(-4) on the zero section. Thus $H^0(f^\ast T)$ is indeed 3-dimensional, as we expected. However, the Euler characteristic of $f^\ast T$ is not 3 but 0, which means that the "expected dimension" is zero.
The meaning of expected dimension is rather vague. Roughly speaking, it is the dimension of the moduli space for a "generic" choice of deformation. The trouble is that such a deformation might not actually exist. Nevertheless, we can still pretend that a generic deformation does exist and, if the expected dimension is zero, compute the number of curves that it "should" have.
What makes this possible is the obstruction bundle E on M. Any deformation of X gives rise to a section of E and the vanishing locus of this section is the collection of curves that can be deformed to first order along with X. Even though a generic deformation might not exist, the obstruction bundle does still exist, and we can make sense of the vanishing locus of a generic section by taking the top Chern class.
In our situation, the (fiber of the) obstruction bundle is $H^1(f^\ast T)$. Since O(2) does not contribute to H^1, the obstruction bundle is $R^1 p_\ast f^\ast O_{P^1}(-4) = R^1 p_\ast O_{P^3 \times P^1}(-4, -4)$ where $p : P^3 \times P^1 \rightarrow P^3$ is the projection. By the projection formula, this is $O(-4)^{\oplus 3}$ and the top Chern class is -64. This is the "Gromov--Witten invariant" of maps from P^1 to $O_{P^1}(-4)$.
Unfortunately, I don't have anything to say about what this -64 means...
Best Answer
You say:
"This issue is addressed directly by Ionel-Parker in their paper on relative GW-invarianst and also by Li-Ruan in their virtual neighborhood construction. Both come to the conclusion, that after a generic choice of almost complex structure, such behavior can be ruled out."
but I don't think that you have correctly characterized their solution to the problem. You do not get to dismiss the behavior --- instead you deal with it in a way that maintains transversality by caveat. In order to explain, it is helpful to compare the way the problem is handled in algebraic geometry and then compare it with what is done in symplectic geometry.
In algebraic geometry, if you want to have a proper moduli space of relative stable maps to $(X,D)$ and maintain the condition that components of the map are not mapped into the divisor $D$, you are forced to allow the target $(X,D)$ to degenerate a singular target $(X,D)\cup\_D F$ consisting of the union of $X$ and $F=\mathbb{P}(\mathcal{O}\oplus N\_{D/X})$ glued along the zero section of $F$ and more generally, you need to allow a finite chain of copies of $F$ glued successively along the $0$ and $\infty$ sections: $X\cup_D F\cup_D\cdots\cup_D F$.
Although it is perhaps a little hidden, this same phenomenon is forced on you in the analytic setting of Li-Ruan and Parker-Ionel. Instead of the singular target considered above, they use a target of $X/D$ with a metric that looks like an
infinitevery, very long tube in the neighborhood of $D$. When they analyze the limiting behavior of maps to such a space, they see a similar phenomenon cropping up: the energy of a map to the tube partitions itself into sections so that in the limit, you can view it as given by a collection of maps to a bunch of long tubes, each of which give you a map to $F$ (the point being that the infinite tube is conformally equivalent to $F$ minus the 0 and $\infty$ sections).There is a picture of this on the top of page 5 of this paper of Parker-Ionel:
http://arxiv.org/pdf/math/0010217
They are actually illustrating the gluing of relative invariants (so that manifolds appear on both sides of the tube instead of just one side), but you can see the appearance of multiple copies of $F$.
So to answer the question: there is no oversight in those Parker-Ionel, Li-Ruan papers. The phenomenon that McDuff is illustrating is a fact of life for relative invariants, even if it is manifested a little differently in the analytic setting.