While not formed to be one, de facto you might consider the Irish Brigade of the Civil War.

Formed as a core around 3 NY Regiments (63rd, 69th, 88th), and at times a MA or PA Regiment. Being recruited almost exclusively from the urban poor recent Irish immigrants the Brigade had an overtly Catholic identity at odd with the Protestant majority. Indeed the Brigade and Regiments funded their own Catholic priests for the men, the most famous being Father Corby who granted the Brigade General Absolution before the 2nd day at Gettysburg, and who would alter serve as President of Notre Dame.