  We present cosmological constraints for the flat $\Lambda$CDM model,
including the sum of neutrino masses, by performing a multi-probe analysis of a
total of 13 tomographic auto- and cross-angular power spectra. This is achieved
by combining, at map level, the latest primary CMB and CMB-lensing measurements
from the Planck 2018 data release, as well as spectroscopic galaxy samples from
BOSS DR12, and the latest Kilo-Degree Survey (KiDS-1000) tomographic weak
lensing shear data release. Our analysis includes auto- and cross-correlations
as well as calibration parameters for all cosmological probes, thus providing a
self-calibration of the combined data sets. We find a good fit (reduced
$\chi^2$=1.7) for the combined probes with calibration parameters only
moderately different from their nominal value, thus giving a possible
interpretation of the tension between the early- and late-Universe probes. The
resulting value for the structure growth parameter is $S_8 = 0.754 \pm 0.016$
(68\% CL). We also obtain a $\sim$2.3$\sigma$ constraint on the neutrino mass
sum of $\sum m_\nu = 0.51^{+0.21}_{-0.24}$ eV (68\% CL), which is compatible
with current particle physics limits. We perform several tests by fixing the
neutrino mass sum to a low value, considering narrower priors on the
multiplicative bias parameters for cosmic shear, and by fixing all calibration
parameters to their expected values. These tests result in worse fits compared
to our fiducial run, especially for the case when all calibration parameters
are fixed. This latter test also yields a lower upper limit of the neutrino
mass sum. We discuss how the interplay between the cosmological and calibration
parameters impact the $S_8$-tension and the constraints on the neutrino mass
sum. [abridged]
