I will divide my understanding into several segments: Hawaiian Recognition, Pan-Asian Diplomacy, and The Letter.

Hawaiian Recognition

The Kingdom of Hawaii was in a unique position during much of the 19th century, as it was one of the few non-Western nations that was generally treated as an equal by Western powers. Hawaiian independence was recognized by Belgium in 1842, by Britain and France in 1843, and by the US in 1844 (Gonschor p. 27). While certain treaties with Britain and France did contain unequal clauses, these were fully revised by 1858 to set Hawaii on equal footing with the Western powers (Gonschor p. 36).

Hawaii continued its rather productive diplomatic policy, establishing over 100 diplomatic missions by 1887 century and even creating its own unequal treaty with Japan (the Hawaiians eventually revised this to equality) (Gonschor p. 37). By 1894, the British legal scholar John Westlake could write:

The international society to which we belong, and of which what we know as international law is the body of rules, comprises - first, all European states . . . secondly, all American states . . . thirdly, a few Christian states in other parts of the world, as the Hawaiian Islands, Liberia and the Orange Free State. The same cannot be said of all Christian states, not for instance of Abyssinia. (Westlake p. 82)

Thus, Hawaii was viewed as a rather special case, as it was generally included within the European family of nations.

Pan-Asian Diplomacy

Hawaii had already established relations with other island nations - Samoa, Tonga, Tahiti, Fiji, among others - earlier in the 19th century. By the accession of King Kalakaua in 1874, the government began to expand its interest to include nations in Asia. In the 1870s the Hawaiian government received Chinese sponsorship to construct a trans-Pacific telegraph cable line, with the Chinese Viceroy Li Hongzhang supporting this endeavor and also establishing a merchant marine company (Gonschor p. 72 - 73). Certain Hawaiian officials also began toying with the idea of encouraging immigration from the East Indies.

In 1881 Kalakaua embarked on his world tour, first heading to San Francisco and from there visiting Japan, China, Siam, and Johor before continuing east. During a meeting with the Emperor Meiji, King Kalakaua explained:

The purpose of my travels has been to promote something that has been on my mind for many years, a league of the countries of Asia. The European countries make it their policy to think only of themselves. They never consider what harm they may cause other countries or what difficulties they may cause other people . . . Consequently, it is imperative for the countries of the East to form a league to maintain the status quo of the East, in this way opposing the European countries. The time for action has come . . . During my trip, I intend to meet with the rulers of China, Siam, India, Persia, and other countries and to discuss with them the advantages and disadvantages of forming a league. Howeer, my country is a tiny cluster of islands and its population is insignificant; it lacks the strength to carry out a great plan . . . (Gonschor p. 77)

Unfortunately, for unknown reasons King Kalakaua never actually visited Persia during the Asian leg of his tour, instead passing from British India straight on to Aden. However, he did visit with Viceroy Li Hongzhang in Tianjin, with King Chulalongkorn in Siam, and with the Maharajah of Johor. The King was especially impressed with the Maharajah, who he found to both physically resemble one of his relatives and whose language had many similarities (Gonschor p. 82).

The Letter

I am not aware of Persia and Hawaii establishing formal treaty relations (and as noted, the King did not visit Persia as intended). However, they did exchange honors and diplomatic communications after the World Tour. I was able to find a transcription of King Kalakaua's letter 1886 here, which reads

We KALAKAUA I., by the Grace of God, of the Hawaiian Islands, King, to His Imperial Majesty Nasser Eddin-Shah-on-Shah of Persia. Great and Good Friend:—

We have read with great pleasure the letter which your imperial Majesty has sent to Us in which, with so many kind expressions of friendship and good will, He accepts the Grand Cross of Our Royal Order of Kamehameha and tenders, as a mark of his sincerity and reciprocality of His statements, to Us, the high distinction of the decoration of the first class of the Lion and the Sun. We hasten to assure your Imperial Majesty of the high satisfaction with which We receive this token of His kindly feeling, toward Our Person and towards Our Country; feelings which We shall ever most heartily reciprocate. (Sai p. 19)

My supposition is that this communication did not continue for terribly long, as King Kalakaua died in 1891 and the Hawaiian Kingdom itself experienced a coup in 1893.

If you are interested in more letters, the Hawaii State Archives include this page where you can find all of the King's handwritten letters that he wrote during his travels, and the University of Hawaii also keeps these typed transcriptions of those letters here.

Sources Directly Cited:

Gonschor, Lorenz. Power in the World: the Hawaiian Kingdom in Oceania. University of Hawai'i Press, 2020.

Sai, David Keanu. Hawaiian Neutrality: From the Crimean Conflict through the Spanish-American War. Paper presented at the University of Cambridge, 2015.

Westlake, John, and L. Oppenheim. The Collected Papers of J W on Public International Law. Cambridge University Press, 1914.