In this paper, we are concerned with the three dimensional Euler equations driven by an additive stochastic forcing. First, we construct global Hölder continuous (stationary) solutions in $C(\mathbb{R};C^{\vartheta})$ space for some $\vartheta>0$ via a different method from \cite{LZ24}. Our approach is based on applying stochastic convex integration to the construction of Euler flows in \cite{DelSze13} to derive uniform moment estimates independent of time. Second, for any divergence-free Hölder continuous initial condition, we show the existence of infinitely many global-in-time probabilistically strong and analytically weak solutions in $L^p_{\rm{loc}}([0,\infty);C^{\vartheta'}) \cap C_{\rm{loc}}([0,\infty);H^{-1})$ for all $p\in [1,\infty)$ and some $\vartheta'>0$.