First Mentioned: LIAB Prologue 19
It is impliedLIAB Prologue 19 that Hyatt is not in fact a Royal side. Hyatt is a Regency, but the meaning of this is unclear. In the Real World's history, a regent is established when a new King or Queen is too young to assume the duties of the throne. The regent rules in the monarch's stead until the monarch comes of age. It is not clear how this would function in a world with no children.
It may indicate a once-Royal side whose monarch named a non-Royal Heir Designate before being croaked. This could possibly happen even with one or more Royal Heirs in the side. It is also possible that the side thus run by a non-royal Overlord, or Regent could still pop a Royal Heir, perhaps within a limited number of turns, or a royal heir was already being produced when the Ruler was croaked with a non-royal unit holding the position of Heir Designate at the time. Or possibly, the (Royal) Ruler is under some long-term effect (possibly a spell) which left the Ruler unable or unfit for ruling so the (non-royal) Chief Warlord had to become Regent and assume control.
As the Hyatt Regency has existed for some time, it may be that there is a royal being popped who/whose capital was affected by some offensive use of turnamancy to delay the heir's arrival.
Connection to the 'Zelda' Great Mind?
Hyatt may be the side of 'the member of the Great Minds Who Think Alike who looks like Princess Zelda'. Princess is a title commonly given to a royal heir, not a monarch; the cannonical Zelda is both royalty and a user of magic; and the name Hyatt shares the first two letters with Hyrule.
If this is true, then it is possible that the 'Zelda Caster' is the ruler of Hyatt and placed the entire side under the control of a regent warlord, similar to an individual city being governed by a regent warlord, until the Zelda Caster would return to take full authority again.
Real World References
The regency of Hyatt is a reference to the Hyatt Regency hotel.