  In the Tricolored Euclidean Traveling Salesperson problem, we are given~$k=3$
sets of points in the plane and are looking for disjoint tours, each covering
one of the sets. Arora (1998) famously gave a PTAS based on ``patching'' for
the case $k=1$ and, recently, Dross et al.~(2023) generalized this result
to~$k=2$. Our contribution is a $(5/3+\epsilon)$-approximation algorithm
for~$k=3$ that further generalizes Arora's approach. It is believed that
patching is generally no longer possible for more than two tours. We circumvent
this issue by either applying a conditional patching scheme for three tours or
using an alternative approach based on a weighted solution for $k=2$.
