If the blinds are water marked it is unlikely you will get it out although having said that in your situation I would still try by applying browning treatment and cleaning.I would want to take them down and off site to do this.
What makes you think they are hessian?
Do you know if they are Sanderson blinds?
There are very few cleaners who will attempt cleaning of blinds as the results are often poor and it easy to damage them with re-hanging often a problem, making the effort just not worth while.
The protective layer your refer to is probably a heat reflection film against strong sunlight.