<i>Even if they don&#x27;t sanction it, can they actually stop you from doing as you please with an asteroid</i><p>Well they could fine you &#x2F; put you in Jail here on earth.  The state can&#x27;t physically prevent you from murdering people in most cases, for example, but they can impose penalties when you do it.  A commercial mining venture is unlikely to be successful if the government of its home country says what it is doing is illegal.