Consider a symplectic manifold $(M,\omega)$ and the space of almost complex structures. These are $J:TM\to TM$ with $J^2=-\text{id}$. A given $J$ is **$\omega$-tame** when $\omega(v,Jv)>0$, and $J$ is **$\omega$-compatible** when it is $\omega$-tame and $\omega(J\cdot,J\cdot)=\omega(\cdot,\cdot)$. The set of either such $J$ forms a contractible subspace. Note that in either scenario we can form a Riemannian metric $g_J$ by twisting $\omega$ in some way with $J$.

In practice, I sometimes see $J$ being tame, and more often I see $J$ being compatible, but in either case I am unsure of the restrictions that each imposes. **Is there any context in which it is necessary/helpful to use one condition over the other?** This need not be related to pseudo-holomorphic curves in Floer theory. **Do certain results fail when I relax "compatible" to "tame"?**

A fundamental difference I see (quoted from McDuff-Salamon's big textbook) is that compatible $J$ minimize the energy of a J-holomorphic curve in its homology class, but not necessarily for tame $J$.