Let's go a step back.Consider the steps in the order finding subroutine.
$\frac{s}{r}$ is the phase of the eigenvalue of the "mod" operator in the routine.
What we obtain after the inverse Fourier Transform in phase determination is an
n-bit approximation to $\frac{s}{r}$.
This in base 10 is like a real number $0.232..$ upto some number of digits.What we require is the fraction which represent this number($\frac{s}{r}$) from which we can obtain $r$ for further use in the algorithm.
Suppose you obtain the approximation as $0.234$.A naive way to get the rational representation is write it as follows and cancelling the common factors:
$$\frac{234}{1000} \rightarrow \frac{117}{500}$$ and see that $gcd(117,500) = 1$.
This naive method required you to know the factors of the numerator and the denominator.But since we are in an algorithm that actually finds the factors of a given integer, this naive method is of no avail to us.
Here is where continued fractions come in.
Define the following:
$$x = 0.234$$
$$x_0 = x;a_0 = [x]$$
Now let
$$x_{i+1} = \frac{1}{x_i - a_i} ; a_{i+1} = [x_{i+1}]$$ until $x_i = a_i$ for some $i$.
Observe that this procedure yield the convergents $[a_0;a_1,...]$ for x.For example applying it $x=0.234$ we observe that $a_0 = 0$,$a_1 = 4$,$a_2 = 3$,$a_3 = 1$,$a_4 = 1$,$a_5 = 1$,$a_6 = 10$ and the algorithm stops.
Hence using these convergents we have
$$0.234 = 0 + \frac{1}{4 + \frac{1}{3 + \frac{1}{1...}}}$$Substituting the values on the RHS, we obtain a rational number $\frac{117}{500}$.By this procedure we obtain the same rational number only here we need not know the factors beforehand.
Though in this conjured example, it may seem that cancelling factors would be easy and short, the continued fractions method offers generality as well as speed as the only operations that we are doing is division and addition.
Clearly no one can say that "Bohmian Mechanics is incompatible with relativity". They can say that "no known reconciliation is known to exist" but even that is questionable. Sheldon Goldstein has been working on ways to make Bohmian mechanics relativistic for awhile with varying success https://arxiv.org/abs/1307.1714
As Luke said in the comment this https://arxiv.org/abs/quant-ph/0105040 is another paper where Goldstein tries to make a backwards causal theory.
There is no widely accepted (by physics community as a whole) reconciliation as of yet between Bohmian mechanics and relativity, but you're not alone in thinking, as Goldstein does, that the answer could lie somewhere with backwards causation. This could also fix up the creation and annihilation problems for Bohmian mechanics since the electron positron creation/annihilation could be viewed as simply a particle that swapped temporal directions as opposed to "disappearing" or "appearing" out of nowhere.
So there is potentially plenty of meat on this bone and hardly anyone working on it. You should try to flesh it out yourself if you are equipped with the knowledge. Might be a real revolutionary possibility here that is being left unattended.
Best Answer
You might want to check out my paper Quantum Computing and Hidden Variables, where I showed that, in discrete hidden-variable theories sort of like Bohmian mechanics, computing the entire trajectory of a hidden variable is probably an intractable problem even for a "standard" quantum computer -- and would let us efficiently solve certain problems, like Graph Isomorphism, that are not known to have efficient quantum algorithms. (This result probably extends to Bohmian mechanics itself, but there are messy issues of formalization there.) What makes this surprising is that a quantum computer can easily sample any individual point in the hidden-variable trajectory (just simulate the system up until that point in time, then measure!). So the only source of difficulty lies in the correlations between the hidden-variable values at different times. In the same paper, I also showed that calculating a hidden-variable trajectory still probably wouldn't let you solve NP-complete problems in polynomial-time: all it would do is improve the square-root speedup of Grover's algorithm to a cube-root improvement! Thus, calculating hidden-variable trajectories provides one of the only examples I know of a computational problem that generalizes what a quantum computer can do, but only "slightly."
There seems to have been very little other work at the intersection of quantum computing and Bohmian mechanics. One reason for that is that Bohmian mechanics naturally lives in a continuous Hilbert space of particle positions, whereas quantum computing naturally lives in a finite-dimensional Hilbert space of qubits. A second reason is that, if you take a standard quantum algorithm (like Shor's algorithm) and try to look at the trajectory of a hidden variable while the algorithm is running, you get basically no additional insight. You'll just see the exponentially-large wavefunction "doing all the work," while the hidden variable bounces around as an almost comically irrelevant-looking fluff on top of it.