  Recent progress in very high energy (VHE, E >100 GeV) $\gamma$-ray
observations, together with advances in the extragalactic background light
(EBL) modelling, allows to search for new phenomena such as $\gamma$-axion-like
particle ($\gamma \rightarrow$ ALP) oscillation and to explore the
extragalactic magnetic field (EGMF) strength and structure. These studies are
usually performed by searching for some deviation from the so-called
absorption-only model, that accounts for only primary photon absorption on the
EBL and adiabatic losses. In fact, there exist several indications that the
absorption-only model is incomplete. We present and discuss the intergalactic
electromagnetic cascade model (IECM) --- the simplest model that allows to
coherently explain all known anomalies. This model has a number of robust
signatures that could be searched for with present and future instruments. The
IECM model may serve as a new background template, allowing to make future
searches for $\gamma \rightarrow$ ALP oscillation more robust. A detailed
account of our calculations is available in astro-ph/1609.01013v2 (A\&A,in
print).
