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.
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