Consider a symplectic manifold $(M,\omega)$ and the space of almost complex structures (a.c.s. for short). These are $J:TM\to TM$ with $J^2=-\text{id}$. An a.c.s. $J$ 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 other times (more often) see it being compatible, but I am unsure of the restrictions that each impose. **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 is with the Levi-Civita connection $\nabla$ associated to the metric $g_J$. Here a compatible or tame $J$ is not necessarily parallel ($\nabla J\ne 0$) under $\nabla$. It *is* preserved under the modified connection $\tilde{\nabla}_XY=\nabla_XY-\frac{1}{2}J(\nabla_XJ)Y-\frac{1}{4}(\nabla_{JY}J+J\nabla_YJ)X$, and its torsion is a multiple of the Nijunheis tensor. Now when $J$ is compatible then this connection preserves the metric, but not when $J$ is simply tame. I don't really know if this affects its use.

Even more different (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 $J$.