  We perform a joint analysis of the counts of redMaPPer clusters selected from
the Dark Energy Survey (DES) Y1 data and multi-wavelength follow-up data
collected within the 2500 deg$^2$ South Pole Telescope (SPT) SZ survey. The SPT
follow-up data, calibrating the richness--mass relation of the optically
selected redMaPPer catalog, enable the cosmological exploitation of the DES
cluster abundance data. To explore possible systematics related to the modeling
of projection effects, we consider two calibrations of the observational
scatter on richness estimates: a simple Gaussian model which account only for
the background contamination (BKG), and a model which further includes
contamination and incompleteness due to projection effects (PRJ). Assuming
either a $\Lambda$CDM+$\sum m_\nu$ or $w$CDM+$\sum m_\nu$ cosmology, and for
both scatter models, we derive cosmological constraints consistent with
multiple cosmological probes of the low and high redshift Universe, and in
particular with the SPT cluster abundance data. This result demonstrates that
the DES Y1 and SPT cluster counts provide consistent cosmological constraints,
if the same mass calibration data set is adopted. It thus supports the
conclusion of the DES Y1 cluster cosmology analysis which interprets the
tension observed with other cosmological probes in terms of systematics
affecting the stacked weak lensing analysis of optically--selected
low--richness clusters. Finally, we analyse the first combined optically-SZ
selected cluster catalogue obtained by including the SPT sample above the
maximum redshift probed by the DES Y1 redMaPPer sample. Besides providing a
mild improvement of the cosmological constraints, this data combination serves
as a stricter test of our scatter models: the PRJ model, providing scaling
relations consistent between the two abundance and multi-wavelength follow-up
data, is favored over the BKG model.
