Does a control strain transformed only with the background vector also form granules?

1-Check the strains when they are transformed with the empty plasmid as well as GFp plasmid,if it forms granules or not,

2-Check if your protein is being expressed or not. Also if it is intracellular or extra cellular?

3-You also should use SOC medium or TB medium which already have glucose in ther composition which controls the leaky expression if your protein was toxic and may affect the Rosetta cells because they are already sensitive.

Hello Martin

Other members of  gene family along with empty pET28a vector grow nicely and does not form any granules after induction 

Heterologous expression systems based on promoters inducible with isopropyl-β-D-1-thiogalactopyranoside (IPTG), e.g., Escherichia coli BL21(DE3) and cognate LacIQ/P lacUV5 -T7 vectors, are commonly used for production of recombinant proteins and metabolic pathways. The applicability of such cell factories is limited by the complex physiological burden imposed by overexpression of the exogenous genes during a bioprocess. This burden originates from a combination of stresses that may include competition for the expression machinery, side-reactions due to the activity of the recombinant proteins, or the toxicity of their substrates, products and intermediates. However, the physiological impact of IPTG-induced conditional expression on the recombinant host under such harsh conditions is often overlooked.IPTG is not an innocuous inducer; instead, it exacerbates the toxicity of haloalkane substrate and causes appreciable damage to the E. coli BL21(DE3) host, which is already bearing a metabolic burden due to its content of plasmids carrying the genes of the synthetic metabolic pathway. The concentration of IPTG can be effectively tuned to mitigate this negative effect. Importantly,  induction with lactose, the natural inducer of P lac , dramatically lightens the burden without reducing the efficiency of the synthetic TCP degradation pathway. This suggests that lactose may be a better inducer than IPTG for the expression of heterologous pathways in E. coli BL21(DE3)

https://microbialcellfactories.biomedcentral.com/articles/10.1186/s12934-015-0393-3

If expression of other members of the same family and/or induction in the presence of an empty vector does not lead to granulation, the simplest explanation I can come up with is that the expression of this particular protein leads to a physiological change in the host that somehow modifies the 'stickyness' of the cells (induction of an adhesin, or a change in the lipid composition of the OM, whatever) and therefore the cells clump or, in your terms, 'granulate'.

Have you checked the culture under a microscope?