  We consider the $Sp(4)$ gauge theory coupled to $N_f=2$ fundamental and
$n_f=3$ antisymmetric flavours of Dirac fermions in four dimensions. This
theory serves as the microscopic origin for composite Higgs models with
$SU(4)/Sp(4)$ coset, supplemented by partial top compositeness. We study
numerically its lattice realisation, and couple the fundamental plaquette
action to Wilson-Dirac fermions in mixed representations, by adopting a
(rational) hybrid Monte Carlo method, to perform non-trivial tests of the
properties of the resulting lattice theory.
  We find evidence of a surface (with boundaries) of first-order bulk phase
transitions in the three-dimensional space of bare parameters (one coupling and
two masses). Explicit evaluation of the Dirac eigenvalues confirms the expected
patterns of global symmetry breaking. After investigating finite volume effects
in the weak-coupling phase of the theory, for the largest available lattice we
study the mass spectra of the lightest spin-0 and spin-1 flavoured mesons
composed of fermions in each representation, and of the lightest half-integer
spin composite particle made of fermions in different representations -- the
chimera baryon. This work sets the stage for future systematical studies of the
non-perturbative dynamics in phenomenologically relevant regions of parameter
space.
