\section{Methods}
\textbf{Graphene growth on Copper.}  Graphene films were grown using low-pressure CVD with flowing H$_2$ and CH$_4$ gases \cite{Zhu:2011}. Before growth, the copper foils were electrochemically polished to improve surface cleanliness and remove oxides. These copper foils were then folded into `packets' \cite{Li:2011}, loaded into a quartz tube and pumped to a base pressure of 2\,mTorr. The copper substrates were heated to 1030\,$^{\circ}$C and H$_2$/CH$_4$ was introduced for 1.5 hours at a total pressure of $\approx$60\,mTorr. The substrates were then quenched by removing them from the hot zone while under vacuum.

\textbf{Single-layer WS$_2$ growth on SiO$_2$/Si.} Synthesis of SL WS$_2$ was performed at ambient pressure in a 2-inch diameter quartz tube furnace on SiO$_2$/Si substrates (275\,nm thickness of SiO$_2$). Prior to use, all SiO$_2$/Si substrates were cleaned in acetone, isopropanol, and Piranha etch then thoroughly rinsed in DI water. At the center of the furnace  a quartz boat containing ~1\,g of WO$_3$ powder was positioned. Two SiO$_2$/Si  wafers were positioned face-down, directly above the oxide precursor. The upstream wafer contained perylene-3,4,9,10-tetracarboxylic acid tetrapotassium salt (PTAS) seeding molecules, while the downstream substrate was untreated. The hexagonal PTAS molecules were carried downstream to the untreated substrate and promoted lateral growth of the TMD materials. A separate quartz boat containing sulfur powder was placed upstream, outside the furnace-heating zone. Pure argon (65\,sccm) was used as the furnace was heated to the target temperature. Upon reaching the target temperature of 825\,$^{\circ}$C, 10\,sccm H$_2$ was added to the Ar flow and maintained throughout the 10\,minute soak and subsequent cooling. 

\textbf{Heterostructure fabrication.} Bulk hBN crystals were exfoliated onto a $n$-doped TiO$_2$ substrate using scotch tape to obtain 10--30\,nm thick flakes. The TiO$_2$  substrate with exfoliated hBN flakes was annealed in ultra-high vacuum (UHV) at 150\,$^{\circ}$C for 15\,minutes to get rid of any unwanted tape residues.  Next, CVD grown single-domain SL WS$_2$ was transferred onto the cleaned hBN flakes using a thin PC film on a PDMS stamp employing a custom-built transfer tool \cite{Katoch:2018,Ulstrup:2016}. This was followed by another annealing cycle of the WS$_2$/hBN heterostructure in UHV at 150\,$^{\circ}$C for 15\,minutes. Separately, one side of the copper foil (with graphene grown on both sides) was spin coated with a PMMA layer. The CVD graphene layer on the backside (without protective PMMA layer) of the copper foil was etched away using reactive ion etching. Next, the PMMA/graphene was floated by etching away the copper using wet chemistry \cite{Patra:2012,Singh:2015}. The PMMA/graphene layer was subsequently transferred onto a clean WS$_2$/hBN stack.  The top PMMA layer was removed by immersing the graphene/WS$_2$/hBN heterostructure in acetone for 15\,minutes followed by a rinse in isopropanol. Lastly, the graphene/WS$_2$/hBN heterostructure was subjected to another annealing cycle in UHV.

\textbf{Spatially-resolved ARPES experiments.} The heterobilayer sample was shipped in air and inserted into the Microscopic and Electronic Structure Observatory (MAESTRO) UHV facility at the Advanced Light Source (ALS) in Berkeley. The sample was annealed at 600~K for 15~minutes prior to ARPES measurements. 

The micro-focused scanning of core level and ARPES spectra presented in Figs. \ref{fig:1}-\ref{fig:2} was carried out in the MAESTRO nanoARPES end-station using an X-ray capillary (Sigray Inc.) with a spatial resolution in this experiment given by $\Delta s = (1.83 \pm 0.03)$~$\mu$m, as explained in the Supplementary Section 5. We used a coarse-motion piezo scanner for spatial maps larger than 30~$\mu$m with a step-size of 1.5~$\mu$m and a fine-motion piezo flexure scanning stage for detailed maps below this range and with a scanning step of 250~nm. Due to the achromaticity of the capillary we were able to perform valence band measurements with variable photon energy in the range 60-160~eV and W 4f, S 2p, B 1s and C 1s core level measurements with a photon energy of 350~eV without apparent loss of spatial resolution. The high efficiency of the capillary enabled acquisition of 4D datasets of the $(E,k,x,y)$-dependent intensity in $\approx$40~minutes. The $(E,k_x,k_y)$-dependent data presented in Fig. \ref{fig:3} were collected using an electron analyzer equipped with custom-made deflectors in the MAESTRO microARPES end-station, such that the sample could be held fixed during BZ mapping. These measurements were acquired with a photon beam focused to a spot size of $\approx$10~$\mu$m using Kirkpatrick-Baez (KB) optics, enabling BZ mapping on a time scale of $\approx$10~minutes for our flakes. The Supplementary Section 1 demonstrates that high quality spectra could be collected from $\theta_1$- and $\theta_2$-domains using the larger beam on the relevant areas identified by the high spatial resolution maps in Figs. \ref{fig:1}-\ref{fig:2}. All data were obtained using hemispherical Scienta R4000 electron analyzers with the energy- and momentum-resolution set at 40~meV and 0.01~\AA$^{-1}$, respectively. The sample was held at room temperature during all the measurements.  



\section{Acknowledgement}
S. U. and J. A. M. acknowledge financial support from VILLUM FONDEN (Grant. No. 15375), from the Danish Council for Independent Research, Natural Sciences under the Sapere Aude program (Grant No. DFF-6108-00409) and from Aarhus University Research Foundation.  R. J. K. is supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD).  S.S. and J.K. acknowledge the start-up funds from Carnegie Mellon University. J.K also acknowledges partial support from the Center for Emergent materials: an NSF MRSEC under award number DMR-1420451.The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This work was supported by IBS-R009-D1. The work at NRL was supported by core programs and the Nanoscience Institute.


\section{Author information}
The authors declare that they have no competing financial interests.
Supplementary Information accompanies this paper.
Correspondence and requests for materials should be addressed to S. U. (ulstrup@phys.au.dk), J. K. (jkatoch@andrew.cmu.edu) or J. A. M. (miwa@phys.au.dk).

