The following separates real and imaginary parts, assuming that x,y are real, leaving no I at the denominator:
expr = Inverse[{{(9 x)/44 - (11 I)/20, (21 y)/47 - (18 I)/
71, -(21/41) - (19 I)/26}, {-(19/67) - (29 I)/33, -(3/4) + (
3 I)/29, -(101/135) - (6 I)/7}, {3/22 - (21 I)/23,
22/39 + (15 I)/26, 1/85 - x (7 I)/46}}][[1, 1]];
{1, I}.FullSimplify@ComplexExpand@ReIm[expr]
