\section*{Methods}

\subsection{Femtosecond electron pulse generation in a transmission electron microscope}
The experimental work was carried out in a commercially available transmission electron microscope (JEOL JEM 2100F) that has been modified to allow for the investigation of ultrafast dynamics in a stroboscopic laser-pump/electron-probe measurement scheme~\cite{Feist2017}. As our electron source, we employ a W/ZrO Schottky emitter ($r=\SI{490}{\nm}$ radius-of-curvature) operated at an extraction voltage of $U_\mathrm{ext} =\SI{2}{\kV}$ and a bias voltage of $U_\mathrm{bias} =-\SI{0.3}{\kV}$. Cooling the emitter just below the continuous Schottky-emission threshold of \SI{1150}{\K} (filament current \SI{1.6}{\A}), we generate ultrashort electron pulses via linear photoemission by focusing laser pulses (\SI{160}{\fs} pulse duration, \SI{515}{\nm} central wavelength, \SI{600}{\kHz} repetition rate) onto the apex of the nanotip. While we estimate that every electron pulse initially consists of up to a few hundred electrons~\cite{Bach2019}, apertures in the electro-optical beam path limit the transmitted beam to electrons that were generated close to the optical axis, resulting in average transmitted bunch charges of below one electron per pulse. Subsequent acceleration to \SI{200}{\keV} energy and coupling into the microscope column enable a pulse characterisation in real- and reciprocal space, spectral pulse properties are determined by an imaging energy filter (CEFID, CEOS GmbH).

\subsection{Event-driven photoelectron detection}
The photoelectron correlations experiments are imaged with a hybrid pixel electron detector that is based on the Timepix3 ASIC (EM CheeTah T3, Amsterdam Scientific Instruments B.V.) and mounted behind the energy filter. The camera generates a stream of data packages containing the position of electron-activated detector pixels, their time-of-arrival (ToA), which are digitized with \SI{1.56}{\ns} time bins, and the energy (time-over-threshold, ToT) associated with incident electron events. At a beam voltage of \SI{200}{\kV} every individual electron activates a cluster of pixels with variable size ($N_\mathrm{pixels, avg} \approx$ 8 pixels), shape and energy ($\mathrm{ToT}_\mathrm{avg} \approx$ 280 $\,$ a.u.). 

Single-electron event localisation of the ToT-corrected raw data stream is achieved using the Division of Nanoscopy, M4I, Maastricht University event clustering code~\cite{vanSchayckJ.Paul2021}, which is based on a Hierarchical Density-Based Spatial Clustering (HDBSCAN) in Python3. The algorithm reconstructs the timing and position of individual electrons incident on the detector from the activated pixels. Thereby, individual electrons are distinguished in terms of their ToA, attributing between three and nine neighbouring pixels activated within a time window of \SI{100}{\ns} and a summed ToT ranging from from 200 $\,$ a.u. to 400 $\,$ a.u. to the same cluster (see Ref.~\cite{vanSchayck2020}).

In a second step, the photoelectrons are clustered according to the femtosecond laser pulse that generated them. The temporal resolution of the detector (\SI{1.56}{\ns}) is much faster than the temporal pulse separation given by the laser repetition rate ($\approx \SI{1}{\us}$), but much slower than the temporal splitting of the correlated electrons at the detector ($\approx \SI{1}{\ps}$). Hereby, the electrons arriving at the detector within $ \Delta t_\mathrm{n} =$\SI{50}{\ns} are assigned to a number-class electron state $n=1,2,3,...$ determined by the number of electrons in one laser pulse.

\subsection{Double Laser-pulse electron generation}

For the two-laser-pulse generation described in Fig.~\ref{fig_2}f, a Michelson interferometer splits the incoming laser pulse into two separate pulses. One of the interference arms has a variable optical path length, implemented by a retroreflector mounted on a delay stage. The delay time of the two optical pulses is much lower than the repetition rate of the laser ($\approx \SI{2}{us}$). As a consequence, two photoelectrons generated by two separate laser pulses and two photoelectrons generated by the same pulse are both detected as two-electron events.

As the optical power on the laser tip oscillates for small delay times due to constructive and destructive interference of the laser, the number of generated electrons will strongly vary in this delay regime. Therefore, we selected delays with approximately the same one-electron-state rate ($\pm \sigma / 2$) over the integration time of five seconds.


\section*{Acknowledgements}
We thank the members of the G\"{o}ttingen UTEM team for constant support and useful discussions.

\noindent\textbf{Funding Information:}
The work at the Göttingen UTEM Lab was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through  432680300/SFB\,1456 (project C01) and the Gottfried Wilhelm Leibniz program.

\noindent\textbf{Author contribution:}
R.H.: Data curation, Formal Analysis, Investigation, Resources, Software, Visualization, Writing - original draft, Writing – review $\&$ editing. 
A.F.: Conceptualization, Formal Analysis, Investigation, Resources, Software, Supervision, Writing - original draft, Writing – review $\&$ editing. 
T.D.: Investigation (supportive), Resources, Software, Visualization, Writing – review $\&$ editing. 
M.M.: Investigation (supportive) Resources, Software,  Writing – review $\&$ editing. 
S.V.Y: Methodology, Software, Writing – review $\&$ editing. 
C.R.: Conceptualization, Formal Analysis, Funding acquisition, Methodology, Project administration, Software, Supervision, Visualization, Writing original draft, Writing – review $\&$ editing. 



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