Let $X$ and $X'$ be birational holomorphic symplectic manifolds. Then the birational morphism between them identifies $H^2(X)$ with $H^2(Y)$. The period space of $X$ is defined to be a subset of $\mathbb P(H^2(X,\mathbb C))$ define as
$$Per(X) = \{\sigma \in H^2(X,\mathbb C)\mid q(\sigma) = 0, q(\sigma,\bar\sigma)>0\}$$,
where $q$ is the Beauville-Bogomolov-Fujiki form. We have a period map $$ p\colon Def(X)\to Per(X) $$ sending the holomorphic symplectic form on a deformation $X_t$ to its cohomology class. It is a local isomorphism by the local Torelli theorem. Choose a class $\alpha\in H^{1,1}(X)$ and consider the complex plane $F(\alpha) = \mathbb P(\langle\sigma,\bar\sigma, \alpha\rangle)$ inside $\mathbb P(H^2(X,\mathbb C))$. Let $T(\alpha)$ be the intersection of $F(\alpha)$ with $Per(X)$, the twistor line corresponding to $\alpha$.
There are two deformation spaces $\mathcal X\to T(\alpha)$ and $\mathcal X'\to T(\alpha)$ such that $\mathcal X_0 = X$ and $\mathcal X'_0 = X'$. For all $t\in T$, the deformations $X_t$ and $X'_t$ correspond to the same point in the period space, and hence are birational by the Global Torelli theorem.
Is there a way to show that $X_t$ and $X'_t$ are birational without referring to the Global Torelli theorem, at least on some small open subset of $T(\alpha)$?
