(^This picture basically would describe lams.)
Oh god, this is a hard ship to explain. So I think it is canon in the fandom because Lin Manuel Miranda hinted at it in the play and also because he ships it. He said that in a tweet. So in fandom lams is real.
Real life is a lot more complicated. And I mean a lot. First of if they were in a relationship they would have to be very secretive. Same sex relationships were illegal in both Britain and America and punishable by death. The would be a small chance they would survive if found out though. But they would have to flee to a distant country where they were not known and change their identities. And it would be very hard to keep a secret like that during the war. So that alone means it would be very unlikely that if they had a relationship it would be anything more than feelings and love letters. That's what would happen if they had a relationship.
Now did they have a relationship? It is really hard to say. Our only evidence supporting it is the letters shared between the two. And even then we don't know the whole letters because John Church Hamilton intentionally omitted phrases in them saying, "I shall not publish the whole of this." The letter that suggests a relationship the most was sent from Alexander Hamilton to John Laurens.
[Middlebrook, New Jersey, April, 1779]
Cold in my professions, warm in my friendships, I wish, my Dear Laurens, it might be in my power, by action rather than words, to convince you that I love you. I shall only tell you that 'till you bade us Adieu, I hardly knew the value you had taught my heart to set upon you. Indeed, my friend, it was not well done. You know the opinion I entertain of mankind, and how much it is my desire to preserve myself free from particular attachments, and to keep my happiness independent on the caprice of others. You should not have taken advantage of my sensibility to steal into my affections without my consent. But as you have done it and as we are generally indulgent to those we love, I shall not scruple to pardon the fraud you have committed, on condition that for my sake, if not for your own, you will always continue to merit the partiality, which you have so artfully instilled into me. (...) And Now my Dear as we are upon the subject of wife, I empower and command you to get me one in Carolina. (...)
If you should not readily meet with a lady that you think answers my description you can only advertise in the public papers and doubtless you will hear of many competitors for most of the qualifications required, who will be glad to become candidates for such a prize as I am. To excite their emulation, it will be necessary for you to give an account of the lover—his size, make, quality of mind and body, achievements, expectations, fortune, &c. In drawing my picture, you will no doubt be civil to your friend; mind you do justice to the length of my nose and don't forget, that I ⟨– – – – –⟩. After reviewing what I hI have gratified my feelings, by lengthening out the only kind of intercourse now in my power with my friend. Adieu
Yours.
A HamiltonIn the part where Alexander takes about the length of his nose approximately five words are illegible due to mutilation of the original manuscript (the words have actually been cut out). The deleted words were certainly explicitly sexual, as Hamilton's reference to "the length of my nose" was clearly a joking allusion to the length of his man body part. This suggests that John Laurens has seen it before.
The letter leaves questions in the air of if they were actually in a relationship. No one really knows. A lot of people say that this letter proves it's existence but they were probably writing to each other like each day might be their last and could have just been close friends. And also shippers or just regular people tend to see romance in deep friendships. Those letters could have hit differently then they do now. Alexander Hamilton says in many letters that he loves John Laurens. Here's an example. This is a letter from Alexander Hamilton to Marquis de LaFayette.
November 3, 1782:
Poor Laurens; he has fallen a sacrifice to his ardor in a trifling skirmish in South Carolina. You know how truly I loved him and will judge how much I regret him.You could think that if Hamilton loved Laurens romantically why would he tell someone else. On of Alexander Hamilton's grandchildren would call Hamilton, LaFayette, and Laurens, the 'gay trio' and the three were close. Then from Alexander Hamilton to Nathanael Greene.
October 12, 1782:
I feel the deepest affliction at the news we have just received of the loss of our dear and inestimable friend Laurens. His career of virtue is at an end. How strangely are human affairs conducted, that so many excellent qualities could not ensure a more happy fate? The world will feel the loss of a man who has left few like him behind, and America of a citizen whose heart realized that patriotism of which others only talk. I feel the loss of a friend I truly and most tenderly loved, and one of a very small number.So again he says that he loved Laurens. But the striking part of these two letters was that they were the only times he wrote letters about Laurens' death. 7 sentences in total. That is practically nothing for Hamilton. Very little for supposedly loving him. The only other time something similar to this happened was when his son, Philip Hamilton died. He took months to write to people about it and wrote very little.
Even Alexander Hamilton's last written words to John Laurens was, "Yrs for ever."
They were definitely extremely close but here's the thing. If people if that time were gay or bisexual we would probably never know.
So they could have been in a relationship. It is explicitly stated that Alexander Hamilton loved John Laurens and they were super close. It's probable that they were in some sort of relationship. Sadly we will never know for sure.
(I personally ship them.)
Also you can read this article it's short but is lams related and interesting. It is not bad. https://aminoapps.com/c/hamilton/page/blog/all-the-sexual-things-hamilton-said-to-laurens/r8WL_EdUeu0j3VqaegZRqkLgXE0d6MlmRE

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Hamilton Ships
Fiksi PenggemarThis is a book discussing all of the Hamilton ships and there probability both in history and in fandom. And also explaining them.