According to [this search][1] in $\pi$-base, the *sigma product of incountably many copies of* $\mathbb{R}$ (no longer in pi-Base as of 2023 March) and the [*boolean product topology on* $\mathbb{R}^{\omega}$][3] are connected, $T_2$ and homogeneous. 

However, they are *not* first countable, hence they cannot be homeomorphic to subspaces of $\mathbb{R}^{\omega}$. 

  [1]: https://topology.pi-base.org/spaces?q=~First%20Countable%20%2B%20%24T_2%24%20%2B%20Connected%20%2B%20Homogeneous
  [3]: https://topology.pi-base.org/spaces/S000107