Natural units in Monte Carlo simulations

I'm wondering how, if one sets the Boltzmann constant $$k_B=1$$ in an MCMC simulation used to sample the Boltzmann distribution $$P(x,p) \propto \exp\left(- \frac{\frac{p^2}{2m}+\frac{1}{2}m\omega^2 x^2}{k_B T}\right)$$, the resulting sampled positions and momenta can be converted back to SI units? Or is it that I should consider the temperature to be scaled (and if so how is the conversion made - it would seem that typically there's some energy scale like the dissociation energy in a LJ simulation which allows for the conversion but I don't believe I have one here unless I say 1 Joule is the typical energy scale or something like that)?