LOKE
Indeed it is (probably) the EGT limit. When the Bleed is off there is a greater mass airflow through the engine. To heat this to the same (max EGT) limit we have to pump more fuel into the engine. Thus we get more energy at the turbine which equals more RPMs and thus more thrust.
Actually we should be careful when talking about this kind of situation with a FADEC controlled engine since it will be the FADEC that sets an RPM (or EPR) and accepts whatever EGT that occurs. As the engine gets older the EGT required to achieve rated thrust will increase and thus reduce the EGT 'margin' in the engine.
When the engine is new however there will be a large EGT margin available and so it would be entirely possible (without the FADEC getting in the way, as it were) to achieve the same thrust bleeds on to bleeds off.