This is end game content and completion of the story is mandatory.
Much like the roaming pokémon of gen 2, you're going to need some luck and a time on your hands for this. Most of the tyrants are locked to certain zones, which makes it easier, but 3 of them have total free roam of the world. Search the below areas for each tyrant.
Mulcimer -> Ignitia Maps Including Caves
Eurus -> Drake Isles Maps Not Including Lab Nor Cave
Bolzen -> Cadium Maps Including Amelie's Lab
Nivalis -> Frozen Tundra Maps Including Bunker And Tower And Cave, Including Frozen Lake Cave
Petram -> Desert Including Cave
Tikala -> Khan Woods Including Caves
Byeol -> Palmaya All Maps
Fane -> Haunted Woods, Not Including Cave
Ziegler, Titan, Darine -> Any Wild Encounter All Maps