What a great question! Every intuition suggests that exotic scales/modes would produce exotic harmonies, right? But let me offer three examples of something I've found.
2 The exotic-ness of the scale appears in the melody. The chords are pretty ordinary.
Example #1: Pentatonic melodies are accompanied by chords that use the "missing" notes. Take "Amazing Grace." If we sing it in F, there are only five tones in the melody: F-G-A-C-D. However, the second chord (even when harmonizing the tune in a simple way) changes to Bb at the word "sweet." (Amazing grace, how sweet the sound."
And this is hardly unusual for pentatonic melodies! If the chords matched the "exotic" scale, we could get neither IV nor V. (IV requires scale degree 4 and V requires scale degree 7.)
2 Listen to "Kind of Blue" by Miles Davis. None of the chords are exotic, though we hear modal melodies and (more importantly) unusual key shifting from phrase to phrase or section to section. That's what gives the ground-breaking album its exotic harmonic qualities.
 When "classical" composers like Maurice Ravel dipped their toes (or torsos) into modality, we find the same practice. Typical chords in the accompaniment that support modal melodies. Here's a reduction of a famous moment from his String Quartet.
So, friend, I'd encourage you to relax on which chords you use. Compose your exotic melody first, then find "ordinary" chords to support it. Do that, and you'll be in excellent company!