Context. A number of asymptotic giant branch (AGB) stars are known to exhibit UV excess and/or X-ray emission. These have been considered signposts of a hot white dwarf (WD) companion in a symbiotic system (SySt), but AGB stars are so bright that they easily outshine these companions, thus hampering their detection at optical wavelengths. A recent multi-wavelength investigation on the X-ray-emitting AGB (X-AGB) star Y Gem has confirmed the presence of a WD companion and therefore its SySt nature. Aims. Our goal is to explore the true nature of another X-AGB star, namely CGCS 6306, to investigate whether some objects from this group may in fact be unnoticed SySts with AGB donors. Methods. We analyzed optical spectra and photometric data together with X-ray observations to investigate the properties of the stellar components and accretion process in CGCS 6306. Results. CGCS 6306 is a carbon Mira with a pulsation period of 362 days. Its optical spectrum exhibits the typical saw-shaped features of molecular absorptions in addition to H I and He I recombination and [O I] and [O III] forbidden emission lines. The Hα line profile is broad, which can be interpreted as evidence for an accretion disk. The X-ray spectrum is hard, which is typical of highly extincted hot plasma emission, and the X-ray luminosity is ≈1032 erg s‑1. Conclusions. The detection of high-excitation optical emission lines and the X-ray properties of CGCS 6306 confirm the presence of a WD companion, making it a bona fide δ-type X-SySt. Its X-ray luminosity is comparable to that of Y Gem, the other X-AGB star confirmed to be a SySt, which was found to exhibit a high accretion rate. The lack of suitable information on the UV and blue optical properties of CGCS 6306, however, precludes a definitive estimate of the accretion rate in this system. Since CGCS 6306 is a carbon Mira, our work adds it to the small group of Galactic carbon SySts.