I think this example is just plain out weird. If you make a \sim \text{normal}(\mu_a, \sigma_a), b \sim \text{normal}(\mu_b, \sigma_b) and c \sim \text{normal}(\mu_c, \sigma_c) but also c = a + b, all this does is to say that c has two measures - an induced or pushforward one and a “natural” one -, namely:
\pi_1(c) = \mathcal{N}(c \mid \mu_a + \mu_b, \sqrt{\sigma_a^2 + \sigma_b^2}) and \pi_2(c) =\mathcal{N}(c \mid \mu_c, \sigma_c). Since under the hood you have
target += normal_lpdf(c | a_mu + b_mu, sqrt(square(a_sigma) + square(b_sigma));
target += normal_lpdf(c | c_mu, c_sigma);
it turns out that the resulting induced measure, \pi^\prime(c) = \pi_1(c)\pi_2(c), is also a normal. But this could easily not be the case. In summary, I don’t understand what you mean by “joint” prior.
See this discussion for more details.