Quantum annealing can be compared to simulated annealing (SA), whose
"temperature" parameter plays a similar role to QA's tunneling field
strength. However, in SA the neighborhood stays the same throughout
the search, and the temperature determines the probability of moving
to a state of higher "energy". In QA, the tunneling field strength
determines instead the neighborhood radius, i.e. the mean distance
between the next candidate state and the current candidate state.
In other words, the "tunneling field" and, indeed, the term "quantum" are just metaphors for the finer points of a completely classical (although randomized) algorithm. QA is not inherently connected with actual quantum computers, it just borrows terms from physics to make a point about computational properties.