Note: Not a solution but marked as such to elevate a key reply
Hi everyone 
You can open a task in a new tab by right-clicking on the full-screen button (top right of the task details pane) and choosing open in a new tab or window.
Asana Desktop app:
Asana in browser:
Alternatively on PC, you can either:
- hold Ctrl + left mouse click…
or - middle mouse click…
…on the full screen icon to directly open that task in a new tab!