  Memoryless (Markovian) system-bath interactions are of fundamental interest
in physics. While typically, the absence of memory originates from the
characteristics of the bath, here we demonstrate that it can result from the
system becoming lossy due to the Markovian interaction with a second bath. This
uncovers an interesting interplay between independent baths and suggests that
Markovianity is ``contagious'', i.e., it can be transferred from one bath to
another through the system with which they both interact. We introduce a
Bloch-Redfield-inspired approach that accounts for this distinct origin of
Markovianity and uniquely combines non-Hermitian Hamiltonian formalism with
master equations. This method significantly improves the description of the
interaction between a lossy system (associated with a Lindblad master equation)
and a non-Markovian bath, reducing the computational demands of complex
system-bath setups across various fields.
