EIRP is not the same thing as Transmitter Output Power (TP), the two
different measurements are only nominally related.

Assuming the wire or duct carrying the RF energy from the transmitter has
ZERO REISISTANCE, and no HYSTARHESIS (Z0) effects:

TP (will never exceed) EIRP, assuming the antenna is a single point radiator
(or an omnidirectional dipole for longer frequencies).

Example of almost 0 db directional gain (AM broadcasting):
http://cbc.am/fk.htm
-- my goal here is to have 26 dbi gain, not omidirectional gain.

An example of 16 db directional gain (shortwave broadcasting):
http://cbc.am/cbc.htm
-- (EIRP) gain is spread out over a 40 degree region
-- notice the low (50 kw vs 500 kw) power requirements due to the
directional gain aka EIRP

Classical directional gain in a 30 degree region:
http://cbc.am/rci-bc.htm

With enough radiators arranged in the right order, 26 dbi gain is easily
obtained -- and the need for extremely high power transmitters is reduced.
Directional gain antenna systems make modern broadcasting possible...