= Solution
A real <matrix> is a <Hermitian matrix> exactly when it is a <symmetric matrix>. Since $R_1$ is orthogonal, this is equivalent to $R_1=R_1^{-1}$, or $R_1^2=I$. A generic rotation does not satisfy this; for example $\theta=0$, $\phi=\pi/2$ gives a nonsymmetric quarter-turn. Thus $\boxed{R_1\text{ is not Hermitian in general.}}$
The exceptions are the identity and half-turns, as characterized by <real rotation matrices that are Hermitian>. For this product, a half-turn occurs when either $\theta$ or $\phi$ is an odd multiple of $\pi$; the identity occurs when both are even multiples. For example, the half-turn condition follows from $\operatorname{tr}R_1=(1+\cos\theta)(1+\cos\phi)-1=-1$.
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