  Loss of inputs can be detrimental to the security of quantum position
verification (QPV) protocols, as it may allow attackers to not answer on all
played rounds, but only on those they perform well on. In this work, we study
\textit{loss-tolerant} QPV protocols. We propose a new fully loss-tolerant
protocol QPV$_{\textsf{SWAP}}$, based on the SWAP test, with several desirable
properties. The task of the protocol, which could be implemented using only a
single beam splitter and two detectors, is to estimate the overlap between two
input states. By formulating possible attacks as a semi-definite program (SDP),
we prove full loss tolerance against unentangled attackers restricted to local
operations and classical communication, and show that the attack probability
decays exponentially under parallel repetition of rounds. We show that the
protocol remains secure even if unentangled attackers are allowed to quantum
communicate, making our protocol the first fully loss-tolerant protocol with
this property. A detailed analysis under experimental conditions is conducted,
showing that QPV$_{\textsf{SWAP}}$ remains fairly robust against equipment
errors. We identify a necessary condition for security with errors and simulate
one instance of our protocol with currently realistic experimental parameters,
gathering that an attack success probability of $\leq10^{-6}$ can be achieved
by collecting just a few hundred conclusive protocol rounds.
