The new Spectrum Roentgen Gamma (SRG)/eROSITA X-ray telescope has a superior response to extended soft X-rays in terms of effective area, energy resolution, and field-of-view (FoV). This makes SRG/eROSITA ideal for studying low X-ray surface brightness emission of cosmic filaments. We search for extended X-ray emission between the two nearby galaxy clusters Abell 3667 and Abell 3651 that are separated by a projected transverse distance of ${\sim} 13\,\mathrm{Mpc}$, using data from the SRG/eROSITA All-Sky Survey. Detailed X-ray image analysis of the region between the two galaxy clusters and redshift analysis of sources between them is performed. We carried out thorough surface brightness analysis between the clusters and in their outskirts studying enhanced emission in different directions. The analysis is complemented with an X-ray pointed observation from XMM-Newton, infrared 2MASS data and redshift information from NED. We discover an emission filament beyond the known radio relic in the northwest of A3667 and even beyond three times its virial radius, smoothly connecting to A3651. The X-ray emission in the direction of the filament shows a $30\pm3\,\%$ enhancement with a significance of $11\,\sigma$. The 2MASS map and redshift analysis show an alignment of sources along the filament and make a projection effect unlikely. Taking the redshift progression of sources within the filament into account, its three-dimensional length is estimated to be in the range of $25\,\mathrm{Mpc} - 32\,\mathrm{Mpc}$. Surface brightness analysis in combination with assumptions for ranges of plausible temperatures and metallicities leads to estimates of total flux, gas mass and central baryon overdensity of $F_\mathrm{X}= (7.1^{+2.1}_{-1.0})\times 10^{-12}\,\mathrm{erg s^{-1} cm^{-2}}$, $M_\mathrm{g}=(2.8^{+5.4}_{-1.0})\times 10^{14}\,\mathrm{M_\odot}$ and $\delta_0=220^{+390}_{-65}$, respectively.
