Assume there is a solution $(a_0,\dots,e_0)$. Let $a_0=\min\{a_0,\dots,e_0\}$ without loss. Viewing it as a quadratic in $a$, we get \[ a^2-(b_0 c_0 d_0 e_0)a + b_0^2+c_0^2+d_0^2+e_0^2 + 78=0. \]Then note that if $a_1$ is the second solution, we get \[ a_1+a_0 = b_0c_0d_0e_0 \quad\text{and}\quad a_1 a_0 = b_0^2+c_0^2+d_0^2+e_0^2+78. \]The first equation tells $a_1\in\mathbb{Z}$ whereas the second tells $a_1>0$. Moreover, $a_1>a_0$. By continuing this logic, it is not hard to see that one can generate an infinite 5-tuple of solutions where the sum of entries is strictly increasing. However, I could not manage to find a `primitive solution', hope one of you can nail that part! See below by RagvaloD: $(a_0,\dots,e_0) = (1,1,3,9,10)$ is an initial solution.

$(a,b,c,d,e)=(1,1,3,9,10)$ can be initial solution