Are homogeneous rational functions ratios of homogeneous polynomials?

Question

Suppose that we have a rational function on $\mathbb R^n$ $$ r(x)=\frac{p(x)}{q(x)} $$ where $q$ has no real roots except $0$, which is a root. Suppose that $r$ is zero homogeneous, meaning that $$ r(tx)=r(x)\quad\text{ for all }t>0,\,x\neq 0. $$ We do not assume that $p,\,q\in\mathbb R[x]$ are necessarily homogeneous polynomials. Is it the case that there exist homogenous polynomials $\tilde p,\,\tilde q\in\mathbb R[x]$ such that $$ r(x)=\frac{\tilde p(x)}{\tilde q(x)}? $$ I have no idea if it matters that $q$ has no non-zero real roots. The same might be true if $r$ has any (integer) homogeneity.