A good way to check this sentence is to line up the three events on a timeline and see which tense each one needs.
There are three events here, in the order they actually happened: first, the wallet gets left at home, then the person boards the train, and then the person realizes the wallet is missing. Boarding and realizing happen close together in the story, so both take the simple past tense, "boarded" and "realized." But leaving the wallet happened earlier than both of those, so it needs the past perfect tense, "had left," to mark it as the earlier event.
With this timeline in mind, "left" alone in place of "had left" fails to show that the wallet was left before boarding. Present tense forms like "realize," "board," or "leave" do not fit a story told in the past at all.
Only a version with "boarded," "realized," and "had left," in that order, matches the real sequence of events.
So, the correct answer is As soon as I boarded the train, I realized that I had left my wallet at home.
Reading each of these four sentences aloud, one after another, quickly reveals a small snag in three of them.
Once the snag in each of the other three readings is spotted, one sentence is left that reads naturally from start to finish.
So, the correct answer is As soon as I boarded the train, I realized that I had left my wallet at home.