  We present a comprehensive analysis of the potential sensitivity of the
Electron-Ion Collider (EIC) to charged lepton flavor violation (CLFV) in the
channel $ep\to \tau X$, within the model-independent framework of the Standard
Model Effective Field Theory (SMEFT). We compute the relevant cross sections to
leading order in QCD and electroweak corrections and perform simulations of
signal and SM background events in various $\tau$ decay channels, suggesting
simple cuts to enhance the associated estimated efficiencies. To assess the
discovery potential of the EIC in $\tau$-$e$ transitions, we study the
sensitivity of other probes of this physics across a broad range of energy
scales, from $pp \to e \tau X$ at the Large Hadron Collider to decays of $B$
mesons and $\tau$ leptons, such as $\tau \to e \gamma$, $\tau \to e \ell^+
\ell^-$, and crucially the hadronic modes $\tau \to e Y$ with $Y \in \{ \pi, K,
\pi \pi, K \pi, ...\}$. We find that electroweak dipole and four-fermion
semi-leptonic operators involving light quarks are already strongly constrained
by $\tau$ decays, while operators involving the $c$ and $b$ quarks present more
promising discovery potential for the EIC. An analysis of three models of
leptoquarks confirms the expectations based on the SMEFT results. We also
identify future directions needed to maximize the reach of the EIC in CLFV
searches: these include an optimization of the $\tau$ tagger in hadronic
channels, an exploration of background suppression through tagging $b$ and $c$
jets in the final state, and a global fit by turning on all SMEFT couplings,
which will likely reveal new discovery windows for the EIC.
