Generally a lysogen will protect the bacteria from a super infection by a similar bacteriophage. However does this immunity extend beyond the family level ? Is anyone aware of any example ?
In case that the lyzogenic DNA can be hsoted in the genome of bacteria, a synergestic relation can be enhanced and il benficial to protect bacteria for other invasions. I join for you two publications that can help
And some recent interesting case: Seed, K.D., Lazinski, D.W., Calderwood, S.B., and Camilli, A. (2013). A bacteriophage encodes its own CRISPR/Cas adaptive response to evade host innate immunity. Nature, vol 494, issue 7438, pp 489–491; DOI: 10.1038/nature11927
So far, it seems it is indeed possible in some cases.
While there are likely many different mechanisms that can explain this observation (which is often true) at least a few examples have been worked out. Some lysogenic prophages express proteins that alter the cell surface or change the profile of surface receptors that other phages use to enter the cell. Some lysogen have also been found to harbor restriction/modification systems or to express functions that modify them. So there are a significant variety of mechanisms that some prophages express to reduce infection by other phages that are not specifically repressor immunity.
There are well described cases of such superinfection exclusion/inhibition in the old and recent litterature (see doi: 10.1038/ismej.2016.79. [Epub ahead of print] for instance!) !