naf's comment is correct. This example can be seen as an instance of Mumford's construction of degenerations of abelian varieties into unions of toric varieties. Here is a "purely toric" construction of such a degeneration, where the analytic toroidal structure is explicit (you may already know this construction, but I thought it would be useful to post for completeness):

Take the $\mathbb{Z}^2$-periodic tiling of $\mathbb{R}^2$ by the translates of the triangulation of the unit square $[0,1]\times[0,1]$ into two triangles, by decomposing the square along its diagonal. Let's call this tiling $\mathcal{T}$. Now take $\mathcal{F}:={\rm Cone}(\mathcal{T})$ which results from placing this tiling of $\mathbb{R}^2=\{(x,y,1)\in \mathbb{R}^3\}$ at height $1$ in $\mathbb{R}^3$, then taking the positive span of each triangle in $\mathbb{R}^3$. The result is a decomposition of the positive half-space $\{(x,y,z)\,:\,z>0\}$ into standard affine cones, i.e. cones which are $SL_3(\mathbb{Z})$-equivalent to the positive octant.

Then $\mathcal{F}$ is a normal fan and so the resulting toric variety $\widetilde{X}:=X(\mathcal{F})$ is smooth. Note that it is an complex-analytic toric manifold, but not an algebraic variety, because it has infinitely many toric affine charts. The map $\mathbb{R}^3\to \mathbb{R}$ which projects to the third coordinate induces a map of fans $\mathcal{F}\to \mathcal{G}$ where $\mathcal{G}$ is the fan in $\mathbb{R}$ consisting of the positive axis. Hence, there is a torus-equivariant holomorphic map $\widetilde{X}\to \mathbb{C}$. The fiber $\widetilde{X}_0$ over $0\in \mathbb{C}$ is the toric boundary of $\widetilde{X}$. It consists of an infinite $\mathbb{Z}^2$-periodic quilt of Cremona surfaces as in your question. This is easily seen by observing that the quotient fan at any ray of $\mathcal{F}$ is the fan of the Cremona surface.

Now observe that $\mathbb{Z}^2$ acts by translation on $\mathbb{R}^2$ and this action extends to an action $\mathbb{Z}^2\to SL_3(\mathbb{Z})$ by integral-affine shearing maps on $\mathbb{R}^3$. Furthermore, this action preserves both the fan $\mathcal{F}$ and the projection to $\mathcal{G}$. So we get an action of $\mathbb{Z}^2$ on $\widetilde{X}$ by biholomorphisms, which preserves the map to $\mathbb{C}$. Furthermore, by the above description, the action is easily seen to be free on $\widetilde{X}_0$. It is easily checked that there is an analytic disc $\Delta\subset \mathbb{C}$ around the origin over which the action of $\mathbb{Z}^2$ is free and properly discontinuous. Let's call the inverse image of this disc in $\widetilde{X}$ by $\widetilde{X}^{<1}$.

Now take the quotient $X:=\widetilde{X}^{<1}/(2\mathbb{Z})^2$. It has a holomorphic map $X\to \Delta$, and the central fiber $X_0$ is the quotient of the $\mathbb{Z}^2$-periodic quilt of Cremona surfaces by the action of $(2\mathbb{Z})^2$. So its central fiber has dual complex $S^1\times S^1$. The space $X$ is a smooth complex threefold, because it was the free quotient of a group of biholomorphisms. To understand the general fiber $X_t$ for $0\neq t\in \Delta$, we know that the fiber $\widetilde{X}_t$ was $(\mathbb{C}^*)^2$ and that $\mathbb{Z}^2$ acted on this fiber freely by translations. The quotient is therefore a two-dimensional complex torus (in fact, an abelian variety).

The toric charts on $X$ are clear from this picture, as $X$ was constructed as the free quotient of an open subset of a toric variety.