We present a multi-band search for X-ray, optical and $gamma$-ray emission of the radio binary millisecond pulsar J1836-2354A, hosted in the globular cluster M22. X-ray emission is significantly detected in two Chandra observations, performed in 2005 and 2014, at a luminosity of $sim$2-3$times$10$^{30}$ erg s$^{-1}$, in the 0.5-8 keV energy range. The radio and the X-ray source positions are found consistent within 1$sigma$ error box. No detection is found in archival XMM-Newton and Swift/XRT observations, compatible with the Chandra flux level. The low statistics prevents us to assess if the X-ray source varied between the two observations. The X-ray spectrum is consistent with a power-law of photon index $sim$1.5. We favour as the most probable origin of the X-ray emission an intrabinary shock scenario. We searched for optical and $gamma$-ray counterparts to the radio source using data from Hubble Space Telescope and Fermi-LAT catalogues, respectively. No optical counterpart down to V=25.9 and I=24.7 (3$sigma$) is detected, which suggests a companion mass of 0.1-0.2 $M_odot$. Combined with the low X-ray luminosity, this is consistent with a black widow nature of PSR J1636-2354A. Inspecting the 8-year Fermi-LAT catalogue, we found a $gamma$-ray source, 4FGL J1836.8-2354, with a positional uncertainty consistent with the globular cluster, but not with the radio position of the millisecond pulsar.
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