Here wee use {x, 0.5, r1}, {y, 0.5, r2}, {z, 0.5, r3} to cut the region x*y*z - x - y - z + 2*Cos[45] > 0 and set the PlotRange to some large space PlotRange->{{0, r1 + 1}, {0, r2 + 1}, {0, r3 + 1}} so we get the second plot.

r1 = 3.7; r2 = 1.3; r3 = 3;  intersection = RegionPlot3D[ x*y*z - x - y - z + 2*Cos[45] > 0 , {x, 0.5, r1}, {y, 0.5, r2}, {z,   0.5, r3}, PlotRange -> {{0, r1 + 1}, {0, r2 + 1}, {0, r3 + 1}}]; regs = RegionPlot3D[{0 <= x <= r1 && 0 <= y <= r2 && 0 <= z <= r3, x*y*z - x - y - z + 2*Cos[45] > 0} , {x, 0.5, r1 + 1}, {y, 0.5, r2 + 1}, {z, 0.5, r3 + 1}, PlotStyle -> {Directive[Gray, Opacity[.2]], Directive[Pink, Opacity[.2]]}, Mesh -> None]; GraphicsRow[{Show[regs, intersection], intersection}] 

Here wee use {x, 0.5, r1}, {y, 0.5, r2}, {z, 0.5, r3} to cut the region x*y*z - x - y - z + 2*Cos[45] > 0 and set the PlotRange to some large space PlotRange->{{0, r1 + 1}, {0, r2 + 1}, {0, r3 + 1}} and then we get the second plot intersection.

r1 = 3.7; r2 = 1.3; r3 = 3; intersection = RegionPlot3D[ x*y*z - x - y - z + 2*Cos[45] > 0, {x, 0.5, r1}, {y, 0.5, r2}, {z,    0.5, r3}, PlotRange -> {{0, r1 + 1}, {0, r2 + 1}, {0, r3 + 1}}]  regs = RegionPlot3D[{0 <= x <= r1 && 0 <= y <= r2 && 0 <= z <= r3, x*y*z - x - y - z + 2*Cos[45] > 0}, {x, 0.5, r1 + 1}, {y, 0.5, r2 + 1}, {z, 0.5, r3 + 1}, PlotStyle -> {Directive[Gray, Opacity[.2]], Directive[Pink, Opacity[.2]]}, Mesh -> None]; Show[regs, intersection]