  We address the role of non-local Coulomb correlations and short-range
magnetic fluctuations in the high-temperature phase of Sr$_2$IrO$_4$ within
state-of-the-art spectroscopic and first-principles theoretical methods.
Introducing a novel cluster dynamical mean field scheme, we compute
momentum-resolved spectral functions, which we find to be in excellent
agreement with angle-resolved photoemission spectra. We show that while
short-range antiferromagnetic fluctuations are crucial to account for the
electronic properties of the material even in the high-temperature paramagnetic
phase, long-range magnetic order is not a necessary ingredient of the
insulating state. Upon doping, an exotic metallic state is generated,
exhibiting cuprate-like pseudo-gap spectral properties, for which we propose a
surprisingly simple theoretical mechanism.
