How can I stabilize a berry coulis?
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.