This is very long. I apologize, but the subject is complex and I don't want to leave necessary details out.
Before you mentioned being his step-father, I knew why he was running away: He feels provoked. (I just didn't know what he was being provoked by.) The emotional stress of being told what to do by you is so offensive to him that he has no choice but to get away. You will probably find it much easier in the long run to switch to a very different set of tactics.
First, I need to define "authority"/"authoritarian" in the context I use here. Not every "authoritarian" is an ogre who uses intimidation and corporal punishment to solve every problem. I think most aren't, actually. To me, an example of authoritarianism is the scenario in which the child does something the parent disagrees with, and the parent lectures or otherwise corrects the child, as a superior to a subordinate. The difference in power is a key element, and an undertone that pervades the encounter. In other words, it's not a negotiation. During these interactions, the parent's mind is set up to issue one-way communications, and not to listen. If you remember back to when your parents have scolded you over various things - if you can recall them talking in a way that may have cowed you, and certainly didn't invite a calm, rational debate - that would be one example of the authoritarian conflict resolution approach.
Unless you were there in his life from early childhood, he will not accept you as an authority figure unless you have some exceptional emotional bonding experience. (Unless there's an erupting volcano for you to rescue him and his mom from, this is highly unlikely. Situations you will encounter in everyday life don't carry the right emotional context to rewrite those settings in his mind.) This is not a willful, conscious decision, but the result of hard-wired circuits in his subconscious mind; he has no choice in the matter and is simply acting according to impulses he can't understand or control. He believes them for the same reason that you believe your own impulses, e.g. "I love my wife" or "I hate the Steelers" (supposing you do): your subconscious mind tells you so, and you believe it automatically. As an adult, you can understand this, and modify the process a little from time to time. As a child, the part of his brain that needs to be online to understand this isn't fully online yet and won't be for some time.
He will reject the idea of you exercising authority, and he will hate you if you try to force the issue or get his mom to force the issue. He sees you as an interloper; he has to do what mom says, but who the heck is this guy? Mom married him, but he's just some dude to me. He isn't my dad just because Mom likes him.
If he remembers his real father, trying to exercise authority will only make things so much the worse. To this boy, your exercise of power over him makes you a bully. Believe me, I had friends growing up who constantly fought with their step-parents, who never "got it" that they were operating according to a perceived bond that wasn't there.
The upshot is that you have to give up on the idea of him ever accepting you as an authority figure just because you happen to be married to his mother. That is not a sufficient condition. I don't mean that you shouldn't try to stop him if something terrible is about to happen. I also don't mean that you shouldn't be in charge when your wife is away and you have to look after him. However, authority is not something to rely on.
I would sit down and have a conversation with your wife. Tell her what I told you, that this is mostly about a bond that your arrival was too late to create. The paradigm of "I'm the man of the house and you will obey me" will not work; you cannot earn someone's trust by subjugating them, and what makes him feel compelled obey his mom is not going to work in your case. It can't; the psychological layout of the human mind doesn't support such a conversion at all.
You and your wife will need to come up with a way for you to interact with him that isn't predicated on authority. In a way, this is a good thing. Children raised by authoritarian parents tend to have more shrill, brittle thought patterns. They tend to be more demanding, more unreasonable, more insistent that everyone has to comply with their whims, because this is what authoritarianism teaches. Believe me, I know. It has taken years to revise behaviors I learned in my own upbringing, which was often authoritarian.
Many parents default to authoritarianism because that's easier than trying to explain, and possibly losing an argument that would make them look ineffectual - and also because they allow themselves to become upset, which doesn't exactly motivate a calm, rational conversation. I don't know how many times that happened when I was growing up. Hundreds? Thousands? I could've done without the need to un-learn all those behaviors. I've been working on it for over a decade and I still have a long way to go. Imagine if your son was completely spared this, and entered adulthood with the skill to resolve his issues with words rather than authority and so much stomping around. You'd be doing him a huuuuuge favor.
Despite the difficulty of learning new ways to parent, I would argue that it's worthwhile to figure it out. Children raised by parents who use reason instead of authority learn to see multiple points of view automatically. They learn that being reasonable and thinking about others is a better way to get things done, than to yell and stamp their feet and insist that everything has to be done their way. In fact, if you do this well enough, you may find that there are times he'll come to you instead of his mom. He'll know that you're on more equal footing with him, if not exactly equal. You'll stop him if you have to, but only after trying to reason with him. He'll know that you won't pull rank on him the second things get mildly difficult, which unfortunately is a mistake that many parents have unwittingly copied from their own parents. He'll see you as someone who's willing to leave him the hell alone about things every now and then, to let him make a mistake from time to time without making a big deal about it. I can't begin to tell you how valuable that is.
Every time your son cuts up and you give him that look, you reinforce that sense of being provoked. You trigger his emotions; the running behavior is brought on deck because his subconscious mind knows it'll be needed very soon. Don't trigger him. If you feel yourself getting upset, don't just follow the same pattern you always do. That won't help either of you. Redirect your exasperation into a purposeful plan of action. You'll find it difficult at first, and you'll want to revert to the old behaviors. Don't revert. If you do, figure out what triggered that reversion and remember it the next time you feel yourself about to revert again. Over time, you'll get better at it.
Think in the long term. Use cause-and-effect thinking. The more you do this, the more the old habits of sighing and putting your hands on your hips (or whatever you do that triggers him) will fall by the wayside... and he will appreciate the hell out of you for growing in this way. He won't expect it. It'll be head-and-shoulders above his wildest dreams. His friends who have step parents will gape with jealousy when he tells them about how you treat him.
Now, I'm aware that this may sound somewhat flowery and idealistic. There are hard parts, one of which is figuring out the edge cases. When do you step in and throw your weight around, and when do you reason with him? What's worthwhile to reason with him about, and what doesn't require your correction? If he breaks his toy, you don't have to chastise him; you can just say, "Dang, that really sucks." The toy being broken is already punishment - it does suck. You don't have to make it worse; if he expects to break toys and have you replace them, don't do it. He'll eventually learn from the consequences if they're reasonable, and not something that's forced down his throat.
You will probably mess this up a few times. Don't let yourself slip back into authoritarianism and displays of exasperation. It will be tempting, but if you want to be more than a houseguest in his life, that's the direction you have to go in. You and your wife can figure it out.
Good luck. I wish I'd been raised by people who cared enough to reach out and ask questions like these, rather than just shouting and arm-waving and punishing any time something went wrong.
PS: Whoever downvoted this, will you post a comment about why? If you think I got something wrong, I'd really like to know what it is. I like to learn, so let's talk about it and maybe figure something out, okay?