  We study large-scale inhomogeneous perturbations and instabilities of
interacting dark energy (IDE) models. Past analysis of large-scale perturbative
instabilities, has shown that we can only test IDE models with observational
data when its parameter ranges are either $w_{x}\geq -1$ and $\xi \geq 0,$ or
$w_{x}\leq -1~$ and $~\xi \leq 0$, where $w_{x}$ is the dark energy equation of
state (EoS), and $\xi$ is a coupling parameter governing the strength and
direction of the energy transfer. We show that by adding a factor $(1+w_{x})$
to the background energy transfer, the whole parameter space can be tested
against all the data and thus, the instabilities in such interaction models can
be removed. We test three classes of interaction model using the latest
astronomical data from different sources. Precise constraints are found. Our
analysis shows that a very small but non-zero deviation from pure
$\Lambda$-cosmology is suggested by the observational data while the
no-interaction scenario can be recovered at the 68.3% confidence-level. In
particular, for three IDE models, identified as IDE 1, IDE 2, and IDE 3, the
68.3% CL constraints on the interaction coupling strengths are, $\xi=
0.0360_{-0.0360}^{+0.0091}$ (IDE 1), $\xi= 0.0433_{-0.0433}^{+0.0062}$ (IDE 2),
$\xi= 0.1064_{-0.1064}^{+0.0437}$ (IDE 3). In addition, we find that the dark
energy EoS tends towards the phantom region taking the 68.3% CL constraints,
$w_x= -1.0230_{-0.0257}^{+0.0329}$ (IDE 1), $w_x= -1.0247_{-0.0302}^{+0.0289}$
(IDE 2), and $w_x= -1.0275_{-0.0318}^{+0.0228}$ (IDE 3). However, the
possibility of $w_{x}>-1$ is also not rejected by the astronomical data used
here. Moreover, we find in all IDE models that, as the value of Hubble constant
decreases, the behavior of the dark energy EoS shifts from phantom to
quintessence type with its EoS very close to that a simple cosmological
constant at the present time.
