This is often the hardest and often the most complex and frustrating part of operating from any DX location.
And it can take time!
Luckily many places are signatories of something known as CEPT T/R 61-01, or CEPT for short so if you have a FULL licence (I am from UK so will talk about that) you can often operate with very little issue. The callsign is often DXlocationprefix/YOURHOMECALLSIGN. You cannot operate like this with a novice or intermediate licence (This might change but it was a constant source of confusion for many). Basically, the assorted people in charge just make up the rules as they go along. I was granted SD7B in Sweden but ONLY when on IOTA and ONLY for a specific period. Before that was SM7/M1KTA/P which is a handful.
Several places insist you have to apply for a local callsign based on the fact you have the home licence, or a pass of the exam, again this is pretty well restricted to FULL licencees only. This can result is a shorter callsign with out a prefix being needed. e.g C56KTA for Gambia, instead of C5/M1KTA so it can be something completely different. I obtained a callsign in Mauritius in 2008 on the basis that I had passed the exam for a UK licence application and could use it to apply for a local callsign. It didn’t permit /P operating. Most of the countries that are not signatories to CEPT have different requirements. If you plan on a trip, please make sure you have that licence sorted before hand.
Many of these countries will NOT support remote applications so you might have to think about how you arrange this in the first instance. I used in country legal (lawyers) firms that also dealt with mobile phone licensing as they will probably know who and where to ask.
If you go in person then you might use that trip to scout out possible operating locations (perhaps take a general band RX and travel about where you plan on operating (on HF anyway) and listen to the local noise levels, a site 100 yards further down the beach might be a better option.
Watch out!!! The licence requirements can be different, don’t assume that they are exactly the same as back home as they might not be. CHECK.
Sadly there are several groups that have caused issues for following visitors too, when they operated often albeit for a short contest but caused interference that stopped all later applications. e.g. ZB2 is not a signatory to CEPT T/R 61-01. Therefore all visiting operators must obtain a reciprocal licence from the Gibraltar Regulatory Authority (GRA). A Gibraltar address for the period of the licence will be required. The application form and further information are available here:
https://www.gra.gi/uploads/documents/Communications/Licenses/Terrestrial/402af.pdf
and
https://www.gra.gi/uploads/documents/Communications/Licenses/Terrestrial/402af.pdf
Note that you will not be granted permission for operation on HF. The only bands permitted are 6m, 4m, 2m and 70cm. There is a power limit of 100W e.r.p.and linear amplifiers are not permitted.
What might have worked once doesn’t now 🙁