I agree with Domo, you're situation sounds complex. You would need to gather details of the number of days in each country and then determine the tax consequences, if any, for each country. For example, due to the number of days you are present in Asia and Africa you may well have tax issues to consider in those countries.
From an Irish side of things, you are correct that having been non resident in Ireland for 4 years if you become tax resident again then for 3 years you will be treated as resident but non ordinarily resident. You will be liable to Irish tax on the remittance basis for the 3 years. If you remain non resident you will be taxable on Irish source income only. In both cases, if you have workdays in Ireland there may be an Irish tax charge (and PAYE requirement for your employer).
If you are non resident, non ordinarily resident and have no workdays in Ireland then your employment income should not come within the charge to Irish tax. The Irish Revenue do not require proof that you are paying tax elsewhere. However, as mentioned above, you will need to meet your tax obligations, if any, in those other countries also.