It was shown in [this paper (see formula (2) there)][1] that any normal random variable (r.v.) $Z$ is multiplicatively infinitely divisible; that is, for each natural $k$ there exist iid r.v.'s $W_1,\dots,W_k$ such that $Z$ equals $W_1\cdots W_k$ in distribution; the distribution of $W_1$ is explicitly described. 

From that description, it is easy to get an entire continuous one-parameter family $\big((X_t,Y_t)\big)_{t\in(0,1)}$ of pairs $(X_t,Y_t)$ of nontrivial independent r.v.'s such that $X_tY_t$ has the (say) standard normal distribution for each $t\in(0,1)$.  

  [1]: http://arxiv.org/abs/1803.09838v1