We present the results from a full polarization study carried out with ALMA during the first VLBI campaign, which was conducted in Apr 2017 in the $lambda$3mm and $lambda$1.3mm bands, in concert with the Global mm-VLBI Array (GMVA) and the Event Horizon Telescope (EHT), respectively. We determine the polarization and Faraday properties of all VLBI targets, including Sgr A*, M87, and a dozen radio-loud AGN. We detect high linear polarization fractions (2-15%) and large rotation measures (RM $>10^{3.3}-10^{5.5}$ rad m$^{-2}$). For Sgr A* we report a mean RM of $(-4.2pm0.3) times10^5$ rad m$^{-2}$ at 1.3 mm, consistent with measurements over the past decade, and, for the first time, an RM of $(-2.1pm0.1) times10^5$ rad m$^{-2}$ at 3 mm, suggesting that about half of the Faraday rotation at 1.3 mm may occur between the 3 mm photosphere and the 1.3 mm source. We also report the first unambiguous measurement of RM toward the M87 nucleus at mm wavelengths, which undergoes significant changes in magnitude and sign reversals on a one year time-scale, spanning the range from -1.2 to 0.3 $times,10^5$ rad m$^{-2}$ at 3 mm and -4.1 to 1.5 $times,10^5$ rad m$^{-2}$ at 1.3 mm. Given this time variability, we argue that, unlike the case of Sgr A*, the RM in M87 does not provide an accurate estimate of the mass accretion rate onto the black hole. We put forward a two-component model, comprised of a variable compact region and a static extended region, that can simultaneously explain the polarimetric properties observed by both the EHT and ALMA. These measurements provide critical constraints for the calibration, analysis, and interpretation of simultaneously obtained VLBI data with the EHT and GMVA.
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