The short answer is that a foreign worker can work two jobs in the U.S. These can be 2 full time jobs, 1 full and 1 part time job or 1 full time job with an independent contractor job on the side. But the foreign worker would have to be sponsored for an H1b visa by each employer or company. Two separate job positions means that two separate H1b visa applications must be filed.
Simply calling the second position an "independent contractor" position does not change the analysis. A second H1b visa is still required.
