We study the hyperbolic sine-Gordon model, with a parameter $\be^2 > 0$, and its associated Gibbs dynamics on the two-dimensional torus. By introducing a physical space approach to the Fourier restriction norm method and establishing nonlinear dispersive smoothing for the imaginary multiplicative Gaussian chaos, we construct invariant Gibbs dynamics for the hyperbolic sine-Gordon model beyond the first threshold $\be^2 = 2π$. The deterministic step of our argument hinges on establishing key bilinear estimates, featuring weighted bounds for cone multipliers. Moreover, the probabilistic component involves a careful analysis of the imaginary Gaussian multiplicative chaos and reduces to integrating singularities along space-time light cones. As a by-product of our proof, we identify $\be^2 = 6π$ as a critical threshold for the hyperbolic sine-Gordon model, which is quite surprising given that the associated parabolic model has a critical threshold at $\be^2 =8π$.