I like to think that the Carroburg Greatswords takes their "crowns" from the reiklandcoffers, but the city of Carroburg is clearly in Middenland, they just helped the city to defend itselves from the war with the count of middenland.
So and army from Carroburg could either be "only reiklanders" or middenland with the GS alligned with it, or just the citys own standing army (with their own city collors). The city would have to obey Middenland as it is not a state city and thereby have independent rule. End the end we all have to obey the Emperor!