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:dontknow:...What is EFS/IMEI ?
Every Phone has a partition which contains data relating to your IMEI ,radio settings,data profiles etc.. This in particular is famous with Samsung devices which have the root /EFS folder and which is higly recommended to be backed up before flashing any Custom ROMs..However there is a common misconception that there is no EFS partition for the Nexus since there is no root /EFS Directory and everything is contained in dev/blocks ..
On the Nexus 4 the IMEI info is stored in the three blocks
m9kefs1 (/dev/block/mmcblk0p8)
m9kefs2 (/dev/block/mmcblk0p9)
m9kefs3 (/dev/block/mmcblk0p10)
These blocks contain all the IMEI info pertaining to your device.This data is highly Device specific that means each device has a different data in each block unlike the blocks for recovery,boot which are common for all . So EFS/IMEI of your device is Unique to your phone itself and hence it is necessary to backup your EFS.
WHY BACKUP EFS ?
1) Some points have been covered in the What section itself ( Unique to each device, important data)
2) Sometimes during a flash or even otherwise there is a bad block in your partition which corrupts a part of the EFS/IMEI partition . You are left with nothing but a dead weight you can use as a tablet without calling features.This makes it essential to backup your current partitions to flash the same if ever the same problem occurs with your device.
Thanks for your answers!
unfortunately i dom't have a backup of the efs partition.
My girlfriend has a N5 too, shouldn't it be possible to restore from this?
reflashing the factory image hasn't worked and i tried around 5 times with different approaches.
If restoring from another N5 doesn't work do you think google will replace it?

The Nexus 5 has no built in EFS backup partitions. EFS contains things like your MEID or IMEI... your data profiles.. radio settings. The EFS cannot be replaced... it is unique to your phone only... backing up your OWN EFS is highly recommended.
Update: dees_troy has added EFS backup into TWRP. This does not make my zip obsolete... but the EFS is getting backed up the same way. My zip backs up the 2 EFS partitions and packs them into a flashable zip which you can restore in any recovery. TWRP backs up the EFS just like it does Data, Boot and System.
I want to point out... if using TWRP to backup EFS... try to have an original backup and save that off the device somewhere. Down the road you may end up with lots of EFS backups(if you backup via TWRP each time) and somewhere along the line you could mess your EFS up.. perhaps worse over the course of a month. If you get into the habbit of only saving your most recent EFS backup... that recent EFS backup may already be partially damaged. I'm not saying don't make frequent backups.. just have the very first one tucked away somewhere.
DO NOT RESTORE TWRP EFS BACKUPS TO ANYTHING OTHER THAN THE ORIGINAL PHONE.