When defining the Floer cohomology $HF(L_0,L_1)$ of 2 Lagrangians in a symplectic manifold $(M,\omega)$, one first has to choose some extra data such a 1-parameter family of almost complex structures $(J_t)$. Usually one requires that $J_t$ be compatible with $\omega$, ie that $g(u,v)=\omega(u,J_tv)$ defines a Riemannian metric.
However there is also the related notion of a tame $J$, one such that $\omega(u,Ju)>0$ for all nonzero $u$. My question is:
What goes wrong if we try to use tame but not necessarily compatible $J_t$ to define $HF(L_0,L_1)$?