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Using the triangular-lattice extended Hubbard model as a test system, wecompare $GW$+EDMFT results for the recently proposed self-consistency schemewith causal auxiliary fields to those obtained from the standard implementationwhich identifies the impurity Green's functions with the corresponding locallattice Green's functions. Both for short-ranged and long-ranged interactionswe find similar results, but the causal scheme yields slightly strongercorrelation effects at half-filling. We use the two implementations of$GW$+EDMFT to compute spectral functions and dynamically screened interactionsin the parameter regime relevant for 1$T$-TaS$_2$. We address the questionwhether or not the sample used in a recent photoemission study [Phys. Rev.Lett. 120, 166401 (2018)] was half-filled or hole-doped.

Causal versus local $GW$+EDMFT scheme and application to the triangular-lattice extended Hubbard model