Final answer:
The soft tissue structure that runs through the right and left lobes of the liver is called the falciform ligament.
Step-by-step explanation:
The soft tissue structure that runs through the right and left lobes of the liver is called the falciform ligament. The falciform ligament is a peritoneal fold that separates the right and left lobes of the liver anteriorly. It is actually a remnant of the umbilical vein.