Let $G$ be either a finite group or a Lie group. Let $r_1$ and $r_2$ be two linear representations of $G$. $r_1$ is an ordinary representation but $r_2$ may be projective. Is there a criterion to decide the following question: given $G$, $r_1$, $r_2$, does there exist an ordinary linear representation $R$ such that $r_1 \otimes R$ is isomorphic to $r_2 \otimes R$? As a slight refinement of this question: does there exist an ordinary linear representation $R$ and two ordinary linear representations $T_1$ and $T_2$ where $T_1$ and $T_2$ are each the direct sum of several copies of the trivial representation (possibly a different number of copies in $T_1$ and $T_2$) such that $r_1 \otimes T_1 \otimes R$ is isomorphic to $r_2 \otimes T_2 \otimes R$?
One obvious case when the answer is "no" is that $r_2$ is not an ordinary representation. A second case is if the group is, for example, $SU(2)$ and $r_1$ is a spin $S_1$ representation and $r_2$ is a spin $S_2$ representation with $S_1 > S_2$. Then, $r_1 \otimes R$ will contain higher spin representations than $r_2 \otimes R$ will. Is there a general answer?