  LensMC is a weak lensing shear measurement method developed for Euclid and
Stage-IV surveys. It is based on forward modelling to deal with convolution by
a point spread function with comparable size to many galaxies; sampling the
posterior distribution of galaxy parameters via Markov Chain Monte Carlo; and
marginalisation over nuisance parameters for each of the 1.5 billion galaxies
observed by Euclid. The scientific performance is quantified through
high-fidelity images based on the Euclid Flagship simulations and emulation of
the Euclid VIS images; realistic clustering with a mean surface number density
of 250 arcmin$^{-2}$ ($I_{\rm E}<29.5$) for galaxies, and 6 arcmin$^{-2}$
($I_{\rm E}<26$) for stars; and a diffraction-limited chromatic point spread
function with a full width at half maximum of $0.^{\!\prime\prime}2$ and
spatial variation across the field of view. Objects are measured with a density
of 90 arcmin$^{-2}$ ($I_{\rm E}<26.5$) in 4500 deg$^2$. The total shear bias is
broken down into measurement (our main focus here) and selection effects (which
will be addressed elsewhere). We find: measurement multiplicative and additive
biases of $m_1=(-3.6\pm0.2)\times10^{-3}$, $m_2=(-4.3\pm0.2)\times10^{-3}$,
$c_1=(-1.78\pm0.03)\times10^{-4}$, $c_2=(0.09\pm0.03)\times10^{-4}$; a large
detection bias with a multiplicative component of $1.2\times10^{-2}$ and an
additive component of $-3\times10^{-4}$; and a measurement PSF leakage of
$\alpha_1=(-9\pm3)\times10^{-4}$ and $\alpha_2=(2\pm3)\times10^{-4}$. When
model bias is suppressed, the obtained measurement biases are close to Euclid
requirement and largely dominated by undetected faint galaxies
($-5\times10^{-3}$). Although significant, model bias will be straightforward
to calibrate given the weak sensitivity.
