How can I stabilize a berry coulis? I made a coulis from fresh blackberries by simply using a fork and sieve. I did not cook, neither added sugar to the final juice. The creamy coulis had beautiful color and texture, however it clearly separated into a watery phase and a more granular phase. How can make the coulis a stable substance which works well for decorating a plate?
Xanthan gum if you don't want to modify the flavour.

Or

Boil in heavy sugar syrup if you don't mind sweetening the sauce significantly.

Personally I'd do both, I imagine what you've created it BlackBerry juice without much flavour. I'd first start by boiling the berries in some sugar untill much of the water had evaporated and tasted exactly how I wanted (intense BlackBerry flavour) then if it required use a little xanthan gum to get the right consistency.