External collaboration without doctoral supervisor - is it a fair request to pursue this? I am about two-thirds of the way through my doctoral programme, and am intending to start applying for post-doctoral fellowships within the next few months.

I am fortunate to have already published a first-author publication with my supervisor, and have another under review. I have also brought projects to my supervisor that we are currently undertaking, but will not form part of my thesis.

I wanted to ask for your advice/thoughts on an issue. I have been approached directly by an academic that I greatly respect and have not previously worked with to undertake an analysis on the understanding that I would be first author on a write-up and they would take the last author. While I have a great relationship with my current supervisor, I am keen to try and publish an article without any of my existing/previous collaborators. My reasoning for this is that such a collaboration would enable me to clearly demonstrate my independence as a researcher and my ability to collaborate in any future fellowship applications (or other postdoctoral applications, for that matter).

I wanted to ask you all whether you feel that it is a fair request of my current supervisor that I pursue this external collaboration without them?

Many thanks for any advice you can provide.
Your advisor doesn't own you, so yes, in general you can pursue other research collaborations. 

However, there are situations in which the answer would be otherwise. If you are in a lab science (or similar) in which you are depended on and this would interfere with the work of others, then you have to cover your obligations. 

Since you mention funding as a tag, it may be that you have a grant that requires full time attention to the work of that grant. That might make it more complicated. 

The personality of the advisor might make it inadvisable to do this prior to finishing your degree. It wouldn't be right to forbid you to do this, but it could happen.