How to find all real solution to satisfy this equation without casework or bruteforce?
Since $\tan\alpha+\tan\beta+\tan\gamma=\tan\alpha\tan\beta\tan\gamma$ for $\alpha$, $\beta$, $\gamma$ being three angles in a triangle, pick any triangle and take the tangents of its three angles and you'll have a solution to your equation over the reals: $(\tan\alpha,\tan\beta,\tan\gamma)$.
Start with any set of three numbers $a$, $b$, and $c$, let $s=\sqrt{(a+b+c)/(abc)}$, and let $a'=sa$, $b'=sb$, and $c='sc$. Then $(a',b',c')$ is a solution. The only solution that can't be found by this method is the trivial one, $(0,0,0)$.