Dead Ball or a Foul Ball Out

[Tex]

NFHS, ASA, USSSA rules: Please site rule number.

The field has a temporary fence, about 4 feet high, in the outfield, (left, center, and right field). This fence does not extend through foul territory to either the left field or the right field stands. What is the section of ground that is in foul territory behind this non-existence fence called? Would this area be considered foul ball or a dead ball territory? If a fly ball is caught in this area what is the call; an out where runners can advance or a dead ball strike without any advancement?