fst={{2.22673,0.59492,2.6842},{3.13408,0.16181,1.63544},{1.34213,-0.31185,3.25682},{3.11765,-1.15956,1.2044},{3.21693,-1.46205,-0.21326},{3.25025,1.24013,0.66829},{3.34529,0.95071,-0.68805},{3.32838,-0.43039,-1.13854},{-0.45219,1.37427,3.23023},{0.47229,2.32191,2.63187}}
sec={{9.22673,0.59492,2.6842},{10.1341,0.16181,1.63544},{8.34213,-0.31185,3.25682},{10.1177,-1.15956,1.2044},{10.2169,-1.46205,-0.21326},{10.2503,1.24013,0.66829},{10.3453,0.95071,-0.68805},{10.3284,-0.43039,-1.13854},{6.54781,1.37427,3.23023},{7.47229,2.32191,2.63187}}

perm=Outer[List,fst,sec]

Min[EuclideanDistance @@ perm]