Turns out an ancient paper(*) has the answer.

If z = u₁ + iv₁ and w = u₂ + iv₂, where u₁, u₂, v₁, v₂ ~ N(0,1) (and independent), then the probability density of

r := |wz|

is given by

rK₀(r),

where K₀ denotes the modified Bessel function of the second kind with order 0.

(*) Wells, Anderson, Cell (1962) "The Distribution of the Product of Two Central or Non-Central Chi-Square Variates"