Can only have one per playthrough. However make SAVE once you enter the Taymurian Wilds.
Head North to get infected by the Wood Element, then follow quest marker to cure the disease. Make another save file.
Re-Load orginal save Taymurian Wilds save file.
Now head south to get infected by the wasp. Once infected trophy will pop.
You can now decide which save to continue on as both puppets have pros and cons.