Political splits that have re-emerged in Libya in recent weeks threaten to derail recent progress and hamper crude production increases.

After a brief surge early in the year, Libya’s fortunes appear once more to be on a downward trajectory. The Sharara oil fields, the biggest producers in the country, came onstream briefly in early April before Tripoli’s National Oil Corporation (NOC) again declared force majeure. Gas and condensate output from the Eni-operated Wafa field is...