Generally, phage that produce turbid plaques are temperate phage. The colonies that grow in the middle of the plaque have been infected by the phage, the phage has integrated its nucleic acid into the genome of the host cell and has made that cell resistant to further infection from the phage. The integrated prophage DNA is replicated as part of the host DNA and the phage does not lysis the host. The plaque therefore appears turbid due to the growth of cells containing the lysogenic phage.
Primarily the only solution is to judge it from the nature of plaque if its turbid then it would be temperate if its clear then is lytic phage but be careful sometimes lytic phage can also form such plaques.
Next I would suggest is go for WGS
Then identify some genes in it those code for lysogeny like integrase , repressor proteins .