It seems quite apparent from some of the above posts that many people confuse an EOSID and an emergency turn procedure they are totally different things.
You have to follow an emergency turn procedure in order not to plant yourself into a hill or something (ZRH for example) however the EOSID can be disregarded at Captains discretion if a few factors are met...
My company use to have an EOSID for BCN RWY 20 of climb straight ahead then proceed to VNV to hold...Now why would you on one engine and at a busy time fly towards high terrain when you have miles of sea in front of you where the MSA is reasonable...Ofcourse it was perfectly acceptable in the TO brief to say in the event of...climb straight ahead 3000ft sort the problem out over the water...
The reason the above EOSID was so is because the people that provided our performance insisted that the EOSID ended over a holding pattern...
An emergancy turn however is different as it needs to be followed...