Basal nectaries are not vascularized but phloem bundles reach the tip of the disc.

There is a protective outer layer (a peel or skin) with numerous long, thin strings (the phloem bundles), which run lengthwise between the skin and the edible inner portion.

The precursor to HypSys in tomato is a 146 amino acid polypeptide, exclusively synthesised within the vascular bundles of leaves and petioles associated with parenchyma cells of phloem bundles.

Immunofluorescence microscopy revealed that infection in petioles, stems, and roots of resistant varieties was restricted to one or two phloem bundles and did not spread to neighbouring tissues or to other plant organs.

But I have a feeling that I'll still refer to those "squidgy, stringy bits that run between the skin and the edible portion of a banana" as just that, rather than use the more accurate term "phloem bundles."