My instinct is yes, but I don't know how to formalize it into a proof. I still haven't wrapped my head around spanning trees yet. Any thoughts are appreciated!
The answer is no. Consider this counterexample:
The red edges form the edges of one spanning tree.
The black edges form the edges of another spanning tree.
The complete graph with $5$ verticies provides a counterexample, you can go "around the outside" or "around the star".