With  a  suitably  chosen  bounded  domain  $\Omega\subset  \mathbb R^2$,  we consider  the  Dirichlet   problem  for  a  scalar  reaction-diffusion equation of the form $ u_t = \Delta u + f(u),    t>0,  x  \in \Omega.    $ We show that there is a nonempty set G of functions $f$, that is open in  a $C^1$  topology and  such that  for any  $f\in G$  the following situation  occurs. The  above equation  has two  hyperbolic equilibria $\phi$,  $\psi$   such  that  their  stable   and  unstable  manifolds $W^s(\psi)$ and $W^u(\varphi)$ intersect nontransversally.  This shows that  the transversality  of stable  and unstable  manifolds is  not a generic property in the  class of spatially homogeneous equations.  In contrast,  this property  is  generic if  nonlinearities  of the  form $f=f(x,u)$ are considered (this  was shown by Brunovsky and Polacik).  In  one space dimension, the transversality  occurs always (not only  generically),  as  shown  in  independent  works  of  Henry  and Angenent.