Regular globally hyperbolic maximal anti-de Sitter structures

Let $\Sigma$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce regular globally hyperbolic anti-de Sitter structures on $\Sigma \times \mathbb{R}$ and provide two parameterisations of their deformation space: as an enhanced product of two copies of the Fricke space of $\Sigma$ and as the bundle over the Teichm\"uller space of $\Sigma$ whose fibre consists of meromorphic quadratic differentials with poles of order at most $2$ at the punctures.