"Regimental" is Rennie/reenactor slang for wearing one's kilt without the benefit of underwear. The origin of the term appears to be "regimental style", referring to Scottish Regiments in the United Kingdom army, who wear their kilts this way.
The fact that the UK Armed Forces still have kilt-clad soldiers allows people in the UK to buy wool kilts in army-navy surplus stores, though from what I've read, it is wise to sew a small flap of a less rough fabric in the area that rubs against you -- this technique is taught by quartermasters in the aforementioned military.
In many cases, this is considered the "manly" answer to "What's worn under the kilt?"