We present the first observation of K- and phi absorption within nuclear matter by means of pi- -induced reactions on C and W targets at an incident beam momentum of 1.7 GeV/c studied with HADES at SIS18/GSI. The double ratio (K-/K+)W / (K-/K+)C is found to be 0.319 pm 0.009(stat)+0.014-0.012 (syst) indicating a larger absorption of K- in heavier targets as compared to lighter ones. The measured phi/K- ratios in pi-+C and pi^- +W reactions within the HADES acceptance are found to be equal to $0.55 pm 0.03(stat)+0.06-0.07$ (syst) and to $0.63 pm 0.05(stat)-0.11+0.11$ (syst), respectively. The similar ratios measured in the two different reactions demonstrate for the first time experimentally that the dynamics of the phi meson in nuclear medium is strongly coupled to the K- dynamics. The large difference in the phi production off C and W nuclei is discussed in terms of a strong phiN in-medium coupling.