Who cares about a field manual? It's been approved by soldiers (at most "by order of the secretary of the army"), not by a parliament.
The GC ranks much higher than any FM. It's been signed by the POTUS and ratified by congress. It has the force of a law.

FM 27-10 is pretty much an interpretation guidance booklet, not the rule book itself.


We can stay away from the (usually highly unpleasant, for various reasons) GC in this discussion.
It's not legal to ignore it in a war zone, though.



By the way; the GC III is relevant to the Iraq War 2003-201x:

Art 3. In the case of armed conflict not of an international character occurring in the territory of one of the High Contracting Parties, each Party to the conflict shall be bound to apply, as a minimum, the following
provisions:
(1) Persons taking no active part in the hostilities, including members of armed forces who have laid down their arms and those placed hors de combat by sickness, wounds, detention, or any other cause, shall in all circumstances be treated humanely, without any adverse distinction founded on race, colour, religion or faith, sex, birth or wealth, or any other similar criteria. To this end the following acts are and shall remain prohibited at any time and in any place whatsoever with respect to the above-mentioned persons:
(a) violence to life and person, in particular murder of all kinds, mutilation, cruel treatment and torture;
(...)
(Iraq ratified.)