  We study the two-dimensional stochastic sine-Gordon equation (SSG) in the
hyperbolic setting. In particular, by introducing a suitable time-dependent
renormalization for the relevant imaginary multiplicative Gaussian chaos, we
prove local well-posedness of SSG for any value of a parameter $\beta^2 > 0$ in
the nonlinearity. This exhibits sharp contrast with the parabolic case studied
by Hairer and Shen (2016) and Chandra, Hairer, and Shen (2018), where the
parameter is restricted to the subcritical range: $0 < \beta^2 < 8 \pi$. We
also present a triviality result for the unrenormalized SSG.
