Do two compatible tRNA codons bond together?
Yes, tRNA can form dimers. For example it was shown that E. Coli tRNA GCC forms homodimers, i.e. two identical molecules interact with each other. In this case the dimerization occurs between the anti-codon loops (what was probably meant with UUU and AAA).

References:


Sequence and structure of naturally-occurring tRNA transcripts and site-directed variants are significant barriers to forming oligomers beyond dimers.
(Harold S. Bernhardt, Warren P. Tate, Advances in Bioscience and Biotechnology, 2013, 4, 1-16 ABB)
Anticodon-anticodon interaction induces conformational changes in tRNA: yeast tRNAAsp, a model for tRNA-mRNA recognition.(D Moras, A C Dock, P Dumas, E Westhof, P Romby, J P Ebel, and R Giegé, Proc Natl Acad Sci U S A. 1986 Feb;83(4):932-6)


A naturally occuring mutation, A3243G in human tRNA(Leu(UUR)), causes dimerization via a motif which is not in the anti-codon loop.

Reference:


Wittenhagen LM1, Kelley SO., Nat Struct Biol. 2002 Aug;9(8):586-90., 
Dimerization of a pathogenic human mitochondrial tRNA.