We first present spatially resolved ALMA and VLA continuum observations of the early-M red supergiant Antares to search for the presence of a chromosphere at radio wavelengths. We resolve the free-free emission of the Antares atmosphere at 11 unique wavelengths between 0.7 mm (ALMA band 8) and 10 cm (VLA S band). The projected angular diameter is found to continually increase with increasing wavelength, from a low of 50.7 mas at 0.7 mm up to a diameter of 431 mas at 10 cm, which corresponds to 1.35 and 11.6 times the photospheric angular diameter, respectively. All four ALMA measurements show that the shape of the atmosphere is elongated, with a flattening of 15% at a similar position angle. The disk-averaged gas temperature of the atmosphere initially rises from a value of 2700 K at 1.35 $R_{star}$ (i.e., 0.35 $R_{star}$ above the photosphere) to a peak value of 3800 K at $sim$2.5 $R_{star}$, after which it then more gradually decreases to 1650 K at 11.6 $R_{star}$. The rise in gas temperature between 1.35 $R_{star}$ and $sim$2.5 $R_{star}$ is evidence for a chromospheric temperature rise above the photosphere of a red supergiant. We detect a clear change in the spectral index across the sampled wavelength range, with the flux density ${S_{ u}} propto { u}^{1.42}$ between 0.7 mm and 1.4 cm, which we associate with chromosphere-dominated emission, while the flux density ${S_{ u}} propto { u}^{0.8}$ between 4.3 cm and 10 cm, which we associate with wind-dominated emission. We then perform nonlocal thermal equilibrium modeling of the far-ultraviolet radiation field of another early-M red supergiant, Betelgeuse, and find that an additional hot (i.e., $>7000$ K) chromospheric photoionization component with a much smaller filling factor must also exist throughout the chromospheres of these stars.
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