random way (tumble), or by rotational diffusion [@Saintillan2008; @Saintillan2008a; @Subramanian2009; @Hohenegger2010; @Krishnamurthy2015]. Typically, the critical threshold density is significantly lower in the latter case, and going to zero in the absence of a decorrelation mechanism for individual bacterium orientation. The mean-field kinetic theory has also been extended to systems with steric interactions [@Ezhilan2013; @Ryan2013; @Heidenreich2016; @Reinken2018] and to microswimmers suspended in non-Newtonian fluids [@Bozorgi2013; @Bozorgi2014; @Li2016].

Below the onset of collective motion, the mean-field kinetic theory predicts that the suspension is homogeneous and isotropic, as featureless as a suspension of non-interacting microswimmers. These assumptions are widely used when describing rheological properties of very dilute suspensions [@Hatwalne2004; @Chen2007; @Sokolov2009a; @Saintillan2010; @Underhill2011; @Lopez2015; @Alonso-Matilla2016; @Nambiar2017; @Guo2018; @Nambiar2019; @Liu2019; @Saintillan2018] and enhanced diffusivity of tracer particles [@Wu2000; @Kim2004; @Underhill2008; @Leptos2009; @Dunkel2010; @Ishikawa2010; @Childress2010; @Childress2011; @Kurtuldu2011; @Mino2011; @Mino2013; @Jepson2013; @Pushkin2013; @Pushkin2013jfm; @Morozov2014; @Kasyap2014; @Thiffeault2015; @Patteson2016; @Burkholder2017]. However, recent large-scale Lattice-Boltzmann simulations of dipolar swimmers [@Stenhammar2017; @Bardfalvy2019] revealed the presence of very strong correlations below the onset of collective motion. It was shown that various observables deviate from their mean-field values at any density of microswimmers [@Stenhammar2017], with the deviation diverging in the vicinity of the onset. The origin of such strong correlations can be 