I’m teaching English and find that the textbooks and I often disagree.
I answered yes. I’m from the U.S. Here’s my cite…
Laffs Comedy Club.
I’m guessing if you’re English, it’s Loffs.
To me they rhyme, because I pronounce “laugh” like “laff”
To me they rhyme because I pronounce “staff” like “staugh”
To me they rhyme because I pronounce both like “fizzlethop”.
If you’re from the South it’s Larf; if you’re from the North, it’s Laff.
I moved from the North to the South, and so I’m all over the place. It’s no laughing matter.
j
So what do you and your textbooks say?
Is the southern pronunciation /lɑːf/ or /lɑːrf/? Spelling with an R is confusing to a US English speaker because Britain has a lot of non-rhotic dialects and the US has few.
According to the Cambridge Dictionary, the UK pronunciation of “laugh” is /lɑːf/ (with the vowel of “father”) and the US is /læf/ (with the vowel of “cat”). “Staff” has the same vowel as “laugh” in both countries (like “father” in UK and like “cat” in US). So “laugh” and “staff” should rhyme in any dialect.
Hey, they don’t rhyme if they’re homophones.
It says, “Throatwarbler Mangrove”.
I’m American, I pronounce them as if they rhyme. The British text says that the vowel sound in laugh is there same as in tourist
In one of the two fizzlethops the f is silent, like the second f in cuff.
But it’s the first F in cuff that is silent.
My mistake. In British English, it’s the second f. I’ve lived in the U.S. for over a decade, but I still forget and pronounce it the British way sometimes.
Americans (i.e. North Americans) generally can’t use “larf” as a guide, because they’ll pronounce the “r”. Imagine a Scotsman with “larrrrf”. 
The closest you can bring this to an American really is “loff”.
Lordy - way too rich for me. If I knew how those phonetic descriptions worked, I would have used them. I’m just trying to distinguish between a long vowel and a short one; but you make a good point that my explanation may not mean much to an American.
So: who says the word “laugh”? A comedian. Here are two comedians, one with a northern accent, one with a southern accent, illustrating the “a” in laugh/laughter etc:
UK Southern, long vowel: Stephen Fry (being rather serious):
https://www.youtube.com/watch?v=9fpztV4oMd4 @ 20.45
UK Northern, short vowel: Jason Manford:
https://www.youtube.com/watch?v=h6S96eY_buM @ 3.14 and particularly 4.44
So there you have the two vowel sounds, and I would be interested to know your thoughts. Would an American even be able to easily distinguish something which, to UK ears, is very obvious?
BTW, Jason Manford is taking about his time as a host on an early evening magazine program and reporting on this story: Coventry cat bin dump woman Mary Bale fined for cruelty - BBC News. Whilst there is nothing funny about the news story, his account of having to cope with it is hilarious.
j
PS: I was going to respond with just:* non-rhotic dialect my arse*. But then the voices in my head said to write something serious.
The problem is that no one has exactly textbook pronunciation. It’s like a chart of paint samples with something called “polar white” at one end and “black currant” at the other. At what point does an off-white turn into a pink, red, purple, and finally black?
I pronounce laugh and staff similarly, but what if my “A” vowel comes out highly nasal so it sounds more like “ai” than “aah”?
Long and short, literally, in the sense of how much time the vowel takes up, yes I hear that difference…
My “a as in laugh” is exactly the same as my “a as in cat”. Manford certainly isn’t like that. For him, those are different vowels.
Does “laugh” match a different vowel for Manford than it does for Fry?
I know there are many accents, but is there really an accent where those are the same?
If not a typo, I’d be very wary of this book.
tourist: /ˈtʊə.rɪst/
laugh: /lɑːf/
Nothing remotely similar in those UK pronunciations (from the Cambridge Dictionary).
Is your textbook entitled “English As She Is Spoke” by any chance?