Unfortunately I don't know if the story is true. Certainly Marshal MacMahon first became famous for his successful leadership of French troops during the Crimean War, fought in the 1850s. The French were allies of the Ottoman Turks and MacMahon must have spent time in Turkey as his forces had to travel through the Bosphorus to enter the Black Sea and reach the Crimea. From my research into Ottoman society at this time, it seems that a Muslim woman could not marry a non-Muslim, so if Bosnali Gullu did marry Marhsal MacMahon's son, he would have had to change his religion and convert to Islam. Certainly, there were opportunities for a western doctor in the Ottoman Empire at this time, with the first modern hospitals being established in the 1820s and a Civil Medical School in 1866. Dr Akif's title of bey was given to the administrators of Ottoman sancaks (equivalent to a county or small province), and also to majors and colonels in the late nineteenth-century Ottoman army, which suggests that Dr Akif may have been a military doctor. However, I will have to look into this further to see if there is any real link with Marshal MacMahon.
While researching this story for Mr Atakun however, I came across some wonderful records drawn up in France during the 1770s for Marshal MacMahon's ancestors, who commissioned the research to prove their ancient noble blood for the Court of King Louis XV. To become a noble in France at the time, an aspirant had to be able to prove noble birth in their family back for a least seven generations. The MacMahons paid a famous Irish genealogist then living in France, the Chevalier O'Gorman, to research their family history, which cost the family 12,000 francs (£500), a very considerable sum at the time. O'Gorman's research however, produced rich results.
O'Gorman was able to prove that Marhsal MacMahon's family came from the western peninsula of County Clare, where the MacMahons were Gaelic lords of Corco-Bhaiscinn. Marshal MacMahon's family were a branch of the senior MacMahon family, the Lords of Clonderlaw, the Marshal's family being Lords of Feenish and Rinanagh. Ultimately the Marshal was descended from Mahon, the son of Muirchertach Ua Briain, King of Ireland who died in 1119 AD, who in turn was a great grandson of the famous High-King of Ireland, Brian Boru. According to Chevalier O'Gorman's research the Marshal's family were 'descended, branched off from the House of Clondirala, about the year 1490: this branch was also despoiled of its inheritance about the same time by Cromwell'. For his research into the MacMahon family O'Gorman used medieval Gaelic genealogical manuscripts preserved in Trinity College Dublin, as well as the Book of Lecan, which O'Gorman referred to as 'highly esteemed by the savans of Ireland', and which was 'deposited in the Irish library at Paris', at the time, as well as a collection of Co. Clare genealogies compiled in 1698 'by an historiographer of Thomond'.
Here is Marshal MacMahon's genealogy laid out in full as researched by O'Gorman;
Brian Boru, Tadhg, Tairrdelbhach King of Ireland (d.1086), Muirchertach King of Ireland (d. 1119), Mahon (eldest son), Murroch (Prince of Corcobaiscin, who took the surname MacMahon instead of O'Brien), Diarmaid, Murroch, Donoch, Dairmaid, Rudhriadhe, Donach na glaicé, Teige (eldest son, Lord of Clonderala), Teige, Turloch (d. 1472) married Helen Fitzgerald, daughter of Maurice earl of Kildare, Donach (second son, Lord of Feenish and Rinnanagh, d. 1577), Turloch Lord of Feenish (living 1625) married Jane MacNamara, Brian Lord of Feenish (b. 1568, living 1641) married Margaret daughter of Donach O'Brien of Doagh, Múrtagh Lord of Feenish, married Eleanor O'Nelan, Murroch (d. 1653, dispossessed of his estates by the Cromwellians), married Helen Fitzgerald, Mortogh (d. 1739), married Helen MacSheehy, Patrick (second son, b. 1684, married Margaret O'Sullivan in 1707), had three sons, Marshal MacMahon's ancestor, Jean Baptiste (b. Limerick 1715, emigrated to France, married Charlotte de Belin, naturalised as French 1749, enobled 1750, became Marquis d'Eguilly in 1763). Jean Baptise had two influential brothers - Maurice became a knight of Malta, fought for Charles Stuart in Scotland in 1746, and subsequently served in the Ultonia regiment in the Spanish army. Their third brother became the Catholic bishop of Killaloe.
I think this genealogy is substantially accurate although there are a few very long lived individuals in the sixteenth century. Unfortunately only the more senior MacMahons of west Clare are mentioned in the Annals of the Four Masters. I hope people have enjoyed this blog. I will have to keep an eye out for further sources to see if Marshal MacMahon did indeed have an Ottoman connection, but O'Gorman's research into his family's history is very interesting indeed.