A computational study of the complexes formed by TF3OH (T=C, Si, Ge) with three nitrogen-containing bases NCH, NH3, and imidazole (IM) is carried out at the MP2/aug-cc-pVTZ level. TF3OH can participate in two different types of noncovalent interactions: a hydrogen bond (HB) involving the hydroxyl proton and a tetrel bond (TB) with the tetel atom T. The strength of the HB is largely unaffected by the identity of T while the TB is enhanced as T grows larger. The HB is preferred over the TB for most systems, with the exception of GeF3OH with either NH3 or IM. MgCl2 engages in a Mg...O Magnesium bond (Mg-bond) with the TF3OH O atom, which cooperatively enhances both the HB and TB. The HB strengthening is particularly large for the NH3 or IM bases, and especially for CF3OH, but is slowly reduced as the T atom grows larger. The TB enhancement, on the other hand, behaves in the opposite fashion, accelerating for the larger T atoms. As a bottom line, the Mg-bond generally reinforces and accentuates the preference for the HB or TB that is already present in the dimer. The Mg-bond is also responsible for a proton transfer in the HB configurations with NH3 and IM.