Wiki defines them as:
A synthetic mode is a mode that cannot be derived from the diatonic scale by starting on a different note.
A synthetic chord is a made-up or non-traditional (synthetic) chord (collection of pitches) which cannot be analyzed in terms of traditional harmonic structures,[inconsistent with] such as the triad or seventh chord.
I'm talking about the good old Whole-Tone scale, Octatonic scale, the rest of Messiaen's modes of limited transpositions, Prometheus chord etc.
Examples in use:
Whole-Tone scale - Debussy - Images Book 2 No. 4: Cloches à travers les feuilles
Octatonic scale - Mussorgsky - Boris Godunov: Coronation scene
Prometheus Chord - Scriabin - Prometheus, the Poem of Fire
Farben Chord- Schoenberg - Five pieces for Orchestra: Colours
A few notable composers here include: Rimsky-Korsakov, Mussorgsky, Debussy, Stravisnky, Schoenberg, Bartok, Frank Martin, Scriabin, Roslavets, Messiaen, Takemitsu, Hosokawa...
I will accept in this thread discussions of Polychords such as the Elekrta chord or the Petrushka chord, for the sake of openness.
Notable pioneers: Debussy, Mahler, Strauss, Stravinsky, Ravel.
Wiki again: A polychord consists of two or more (known) chords, one on top of the other.