We prove that the incompressible Euler equations in the plane, without external force, admit two distinct renormalized weak solutions with the same initial vorticity $f$, where $f$ is radial and nonnegative and lies in $(L^1\cap L^q)(\mathbb{R}^2)$ for every $q<\infty$ but is not bounded. One solution is radially symmetric and stationary. The other is $3$-fold rotationally symmetric, is equimeasurable with $f$ at every time, and departs from $f$ in finite time; the two solutions are distinct in $L^\infty_t(L^1\cap L^q)_x$ on $[0,T]\times\mathbb{R}^2$ for every $q\in(4/3,\infty)$ and every sufficiently large $T$.