  It is now well established that the majority of massive stars reside in
multiple systems. However, the effect of multiplicity is not sufficiently
understood, resulting in a plethora of uncertainties about the end stages of
massive-star evolution. In order to investigate these uncertainties, it is
useful to study massive stars just before their demise. Classical Wolf-Rayet
stars represent the final end stages of stars at the upper-mass end. The
multiplicity fraction of these stars was reported to be ${\sim}0.4$ in the
Galaxy but no correction for observational biases has been attempted.
  The aim of this study is to conduct a homogeneous radial-velocity survey of a
magnitude-limited ($V$ $\leq 12$) sample of Galactic Wolf-Rayet stars to derive
their bias-corrected multiplicity properties. The present paper focuses on 12
northern Galactic carbon-rich (WC) Wolf-Rayet stars observable with the 1.2m
Mercator telescope on the island of La Palma.
  We homogeneously measured relative radial velocities (RVs) for carbon-rich
Wolf-Rayet stars using cross-correlation. Variations in the derived RVs were
used to flag binary candidates. We investigated probable orbital configurations
and provide a first correction of observational biases through Monte-Carlo
simulations.
  Of the 12 northern Galactic WC stars in our sample, seven show peak-to-peak
RV variations larger than 10 km s$^{-1}$, which we adopt as our detection
threshold. This results in an observed spectroscopic multiplicity fraction of
0.58 with a binomial error of 0.14. In our campaign, we find a clear lack of
short-period (P~$<~\sim$100\,d), indicating that a large number of Galactic WC
binaries likely reside in long-period systems. Finally, our simulations show
that at the 10% significance level, the intrinsic multiplicity fraction of the
Galactic WC population is at least 0.72.
