There are small "repair kits" available for tarp materials - could carry one for patching small tears. Just get the right kit for your tarp's materials (cuben, silnylon, etc.). Or you could make your own kit and carry that. For large tears your rain gear or your tyvek "floor mat" may suffice temporarily.
Also, tenacious tape has replaced duct tape in my gear. It can even be used as a fire starter (butane-based glue).
"Climb the mountains and get their good tidings.
Nature's peace will flow into you as sunshine flows into trees.
The winds will blow their own freshness into you, and the storms their energy,
while cares will drop away from you like the leaves of Autumn."
— John Muir