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 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, thus, its SySt nature. 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 symbiotic systems with AGB donors. Optical spectra and photometric data, together with X-ray observations, have been analyzed to investigate the properties of the stellar components and accretion process in CGCS 6306. 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 H$\alpha$ line profile is broad, which can be interpreted as evidence for an accretion disk. The X-ray spectrum is hard, typical of highly-extincted hot plasma emission, and the X-ray luminosity is $\approx10^{32}$ erg s$^{-1}$. 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 $\delta$-type X-SySt. Its X-ray luminosity is comparable to that of Y Gem, the other X-AGB 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, it adds to the small group of Galactic carbon SySts.