Be aware that 'Northern Irish' can have political connotations; some may prefer 'Irish' or 'British'. When discussing identity in Ireland, tread carefully because 'Northern Irish' may imply political affiliation.
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Sporting Context
In sports, 'Northern Irish' indicates athletes representing Northern Ireland. The Northern Irish team played exceptionally well last night.
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Geography Specific
Use 'Northern Irish' to specify the region distinct from the Republic of Ireland. The Northern Irish coastline is famous for the Giant's Causeway.