When I talked with students about pseudo-random number generation, I mentioned that you should not blindly use subsequent outputs of a pseudo-random number generator (PRNG) to form tuples as they may not be uniformly random then; a famous example is RANDU.
The natural follow-up question is: what do you do? I can imagine several strategies.
Choose a PRNG that you know works well for $k$-tuples and sample $(\leq k)$-tuples by using subsequent numbers as components.
I assume that $k \leq 5$ is probably covered, but are there any for arbitrary $k$?
Use $k$ different PRNGs (different algorithms and/or different seeds) and draw the $i$th component from the $i$th one.
I seem to remember that mixing PRNGs is not a good idea.
If the domain of your PRNG is big enough, use bits $i,k+i,2k+i, \dots$ for the $i$th component.
Seems sound and is probably the method of choice assumign real $U(0,1)$-numbers, but does not scale given finite resolution/domains in practice.
Use a bijection from the domain of your PRNG to the $k$-dimensional space you want to sample from, e.g. a generalized version of Cantor's pairing function.
The sizes of the resp. domains can become a problem here a well.
None of the approaches seems entirely reasonable. So what is the state of the art? What advice would you give a person who needs to sample uniformly random tuples?