The Supreme Consular Court at Constantinople was based on personal rather than territorial sovereignty. It had power over British subjects in the Ottoman empire. See Geoff Berridge, British Diplomacy in Turkey, 1583 to the Present (Martinus Nijhoff Publishers, 2009) 81-83.
There are three major sources of original material concerning this court.
The fullest appears to be the William Perkins Collection, held by the Middle East Centre, St Antony's College, the University of Oxford. The Centre has a document which describes these materials: http://www.sant.ox.ac.uk/mec/MEChandlists/GB165-0370-William-Perkins-Collection.pdf.
The second source consists of documents held by the National Archives (U.K.). Cases based on records held by the National Archives were selected, edited and transcribed by Peter Bullock, to whom we are very grateful. There is National Archives Case List online.
The British Library's digital newspaper collection contains further references to the Supreme Consular Court at Constantinople. For a large image of the court room, see that collection's copy of The Graphic (London, England), Saturday, April 13, 1878, p. 2.
For commentary on the establishment and operation of these courts, see Newspaper commentary and minor cases, Constantinople.