The beginners run with such issues...
Technically(as the docs can be interpreted), you are employed by the consultant and deputed to Co. where you go to work. This is what Body shopping is.
Cos avoid employment right claim this way.
Since you are leaving and donot wish to press this right, you can discuss with actual Employer Co directly, basis your Original employment intent by them and submit resignation to them and get an acceptance on it.
Formal relieving may not be required then since the tenure has been kind of probation where one can part ways in 24 hrs to 7 days.
Just resign, get it accepted. BUT donot ask for relieving letter in your Resignation Doc. else NEW employer will insisit on it.
Write a separate letter asking for Relieving Letter - if they give it, good, if they donot.. not much to worry. You can tell New Employer that you have been on probation with either party's right to cease by 24 hrs notice.